Differential Geometry and Complex Geometry

Yi-Bing Shen

(Zhejiang University, Email: yibingshen@zju.edu.cn)

Contents

Chapter 1. Riemannian manifolds

Chapter 2. Connections and Curvatures

Chapter 3. Harmonic forms and Hodge theory

Chapter 4. Geodesics and completeness

Chapter 5. Comparison theorems and manifold topology Chapter 6. Complex manifolds Chapter 7. Hermitian and Kähler geometry

Chapter 1. Riemannian Manifolds

1. Classical differential geometry. In R^{3} a piece of smooth surface S may be viewed as the image of a differentiable map D → R^{3} where D is an open domain in uv−plane. The diffeomorhpism ϕ : D → S gives rise to the local coordinates (u, v) on M. The metric on S induced from the Euclidean inner product of R^{3} is called the first fundamental form of S:

ds^{2} = Edu^{2} + 2F dudv + Gdv^{2}.  (1.1)

Gauss’s Theorema Egregium says that the Gauss curvature K of S is determined only by the metric ds^{2} of S. This is a truly remarkable result. If S is compact and closed, then we have the following Gauss-Bonnet formula:

∫_{S} KdA = 2πχ(S),  χ(S) = 2(1 − g),  (1.2)

Typeset by AMS-TEX

∗Partially supported by NNSFC (No. 10271106).

where χ(S) and g are the Euler-Poincarè characteristic and the genus of S, respectively.

In S there exists the local isothermal coordinates (x, y) such that

ds^{2} = λ^{2}(dx^{2} + dy^{2}) = λ^{2}dzdz̄,  (1.3)

where z = x+\sqrt{−1}y. By means of the complex coordinate z, S becomes a complex curve (complex 1-dimension).

The examples of the metrics with constant Gauss curvature K(= 1, 0, −1) are as follows:

(i) ds^{2} = \frac{dx^{2}+dy^{2}}{(1+ \frac{1}{4} (x^{2}+y^{2}))^{2}} = \frac{|dz|^{2}}{(1+ \frac{1}{4} |z|^{2})^{2}} for K = 1.

(ii) ds^{2} = dx^{2} + dy^{2} = |dz|^{2} for K = 0.

(iii) ds^{2} = \frac{4(dx^{2}+dy^{2})}{(1−x^{2}−y^{2})^{2}} = \frac{4|dz|^{2}}{(1−|z|^{2})^{2}} for K = −1.

2. Differentiable manifolds. An n−dimensional differentiable manifold is a Hausdorff topological space M (with countable basis) and a family of diffeomorphisms ϕ_{α} : U_{α} ⊂ M → R^{n} of open sets U_{α} of M into R^{n} such that:

(i) ⋃_{α} U_{α} = M;

(ii) For any pair α, β, with U_{α} ∩ U_{β} = V ≠ ∅, the sets ϕ_{α}(V) and ϕ_{β}(V) are

open in R^{n} and the mapping ϕ _{α} ∘ ϕ_{β}^{−1} are differentiable; (iii) The family {(U_{α}, ϕ_{α})} is maximal relative to the conditions (i) and (ii). The pair (U_{α}, ϕ_{α}) with p ∈ U_{α} is called a coordinate neighborhood at p. We can

attach the coordinate of the point ϕ_{α}(q) ⊂ ϕ_{α}(U_{α}) in R^{n} to the point q ∈ U_{α} ⊂ M. A family {(U_{α}, ϕ_{α})} satisfying (i) and (ii) is called a differentiable structure on M. The condition (iii) is included for purely technical reasons.

Example 1. The Euclidean space R^{n}.

Example 2. The Euclidean sphere S^{n} ⊂ R^{n+1}.

Example 3. The real projective space RP^{n} ∼ S^{n}. S^{n} can be viewed as the twice covering.

Example 4. The Grassmannian G_{k,n} = {k-plane through the origin of R^{n}}. If we choose a basis {e_{1}, ⋯, e_{n}} in R^{n}, then the orthogonal group O(n) acts naturally on R^{n} by matrix multiplication; and since non-singular linear transformations map k-planes to k-planes, we have a map

A : O(n) × G_{k,n} → G_{k,n}.

Observe that if P and Q are k-planes, then there exists a σ ∈ O(n) such that A(σ, P) = Q. Now, let P_{0} be the k-plane spanned by the first k elements e_{1}, ⋯, e_{k}, and let H be the subset of O(n) leaving P_{0} fixed. Then

H = \{( \begin{matrix}α & 0 \\ 0 & β \end{matrix} ) ∈ O(n)|α ∈ O(k), β ∈ O(n − k)\} ,

so H is a closed subgroup of O(n) which we can identify with O(k)×O(n−k). Then the map σ(O(k) × O(n − k)) ↦ A(σ, P_{0}) is a one-to-one map of the quotient space O(n)/O(k)×O(n−k) onto the set G_{k,n}. We make G_{k,n} into a k(n−k)-dimensional manifold by requiring that this map be a diffeomorphism. It is easy to check that this manifold structure on G_{k,n} is independent of the basis chosen for R^{n}. Note that G_{1,n+1} is just RP^{n}.

Example 5. The product manifold M × N. Let {(U_{α}, ϕ_{α})} and {(V_{β}, ψ_{β})} be the differentiable structures on M and N, respectively. Then it is clear that ⋃_{α,β} U_{α}×V_{β} = M ×N. Define the diffeomorphisms ϕ_{α}×ψ_{β} : U_{α}×V_{β} → R^{n}×R^{m} = R^{n+m} as follows:

(ϕ_{α} × ψ_{β})(p, q) = (ϕ_{α}(p), ψ_{β}(q)) ∀(p, q) ∈ U_{α} × V_{β}.

It is easy to see that {(U_{α} × V_{β}, ϕ_{α} × ψ_{β})} is a differentiable structure on M × N.

Let M and N be differentiable manifolds. A map f : M → N is differentiable at p ∈ M if given a coordinate neighborhood (V, ψ) at q = f(p) ∈ N there exists a coordinate neighborhood (U, ϕ) at p such that f(U) ⊂ V and the map f̂ = ψ ∘ f ∘ ϕ^{−1} : ϕ(U) ⊂ R^{n} → ψ(V) ⊂ R^{m} is differentiable at q (see Fig. below).

U ⊂ M ⟶^{f} V ⊂ N
ϕ↓  ↓^{ψ}
R^{n}  ⟶_{f̂=ψ∘f∘ϕ^{−1}} R^{m}

f : M → N is differentiable on M if it is differentiable at all of the points of M. In particular, a differentiable f : M → R is called a differentiable function on M. A (differentiable) curve on M is a differentiable map γ : (a, b) ⊂ R → M.

3. Vector fields and Tensor fields. Let M be a differentiable manifold and p ∈ M a point. Consider a curve γ : (−ε, ε) → M with γ(0) = p. Let C(M) be the set of of functions on M that are differentiable at p. The tangent vector to γ at t = 0 is a map γ′(0) : C(M) → R given by

γ′(0)f = \frac{d(f ∘ γ)}{dt}|_{t=0} ∀f ∈ C(M).

A tangent vector at p ∈ M is the tangent vector at t = 0 of some curve γ : (−ε, ε) → M with γ(0) = p. The set of all tangent vectors to M at p is called the tangent space at p of M, denoted by T M_{p}.

Given a coordinate neighborhood (U, ϕ) at p, we can express the function f and the curve γ in the local coordinates {x_{i}}, 1 ≤ i, j, ⋯ ≤ n, by

f(q) = f(x_{1}, ⋯ x_{n}), (x_{1}, ⋯ x_{n}) = ϕ(q) ∈ ϕ(U),

and

ϕ ∘ γ(t) = (x_{1}(t), ⋯, x_{n}(t)),

respectively. Thus, restricting f to γ, we have

\begin{aligned}γ′(0)f & = \frac{d}{dt}(f ∘ γ)|_{t=0} = \frac{d}{dt}f(x_{1}(t), ⋯, x_{n}(t))|_{t=0} \\ & = ∑_{i} x′_{i}(0)\frac{∂f}{∂x_{i}} = (∑_{i} x′_{i}(0)\frac{∂}{∂x_{i}} |_{t=0}) f. \end{aligned}

It means that γ′(0) can be expressed by

γ′(0) = ∑_{i} x′_{i}(0)\frac{∂}{∂x_{i}} |_{p}.  (1.4)

Observe that \frac{∂}{∂x_{i}} |_{p} is the tangent vector at p of the coordinate curve x_{i} → ϕ^{−1}(0, ⋯, 0, x_{i}, 0, ⋯, 0). The formula (1.4) shows that any tangent vector in T M_{p} can be linearly expressed by \{\frac{∂}{∂x_{i}} |_{p}\}, i.e. they form a basis of T M_{p}. If a coordinate change x̄_{i} = x̄_{i}(x_{1}, ⋯, x_{n}) are given, then the new basis in T M_{p} is determined completely by

\frac{∂}{∂x̄_{i}} = \frac{∂x_{j}}{∂x̄_{i}} \frac{∂}{∂x_{j}} .  (1.5)

Here the Einstein sum convention is used.

As a linear space, the dual space to T M_{p}, whose elements are linear functions fron T M_{p} to R, is called the cotangent space at p of M, denoted by T^{∗}M_{p}. Clearly, \{dx_{i}|_{p}\} is the basis of T^{∗}M_{p}, called the dual basis to \{\frac{∂}{∂x_{i}} |_{p}\}. Similarly, we have

dx̄_{i} = \frac{∂x̄_{i}}{∂x_{j}} dx_{j}.  (1.6)

The set T M := ⋃_{p∈M} T M_{p} = {(p, v)|p ∈ M, v ∈ T M_{p}} may be endowed with a differentiable structure so that T M becomes a differentiable manifold of dimension 2n, which is called the tangent bundle over M. In fact, if {x_{i}} is the local coordinates in a neighborhood U ⊂ M, then any vector v in T M_{p}, p ∈ U, can be expressed as v = ∑_{i} y^{i} \frac{∂}{∂x_{i}}. Taking \{x_{1}, ⋯, x_{n}, y^{1}, ⋯, y^{n}\} as coordinates of (p, v) in T U ⊂ T M, we obtain a differentiable structure for T M.

There exists a natural projection π : T M → M defined by (p, V) ↦ p. Similarly, the set T^{∗}M := ⋃_{p∈M} T^{∗}M_{p} is also a differentiable manifold called the cotangent bundle over M.

By using the tensor product of vector spaces, we can construct the following vector space

T_{s}^{r}M_{p} := T M_{p} ⊗ ⋯ ⊗ T M_{p} ⊗ T^{∗}M_{p} ⊗ ⋯ ⊗ T^{∗}M_{p},

called the (r,s)-tensor space induced from T M_{p}, whose elements are the multilinear maps:T^{∗}M_{p} × ⋯ × T^{∗}M_{p} × T M_{p} × ⋯ × T M_{p} → R. The natural basis of T_{s}^{r}T M_{p} is

\{(\frac{∂}{∂x_{i_{1}}} ⊗ ⋯ ⊗ \frac{∂}{∂x_{i_{r}}} ⊗ dx_{j_{1}} ⊗ ⋯ ⊗ dx_{j_{s}})_{p}\} .

The set T_{s}^{r}M := ⋃_{p∈M} T_{s}^{r}M_{p} is called the (r,s)-tensor bundle over M.

A vector field X on M is a function assigning to each point p ∈ M a vector X_{p} ∈ T M_{p}. In terms of maps, it is a section of T M, X : M → T M, such that π ∘ X = id.. The field is differentiable if the map X : M → T M is differentiable. Clearly, in the local coordinates, X can be expressed by

X = X^{i}(x)\frac{∂}{∂x_{i}} , X^{i} = Xx_{i},

where the components X^{i}(x) of X are the function of x_{j} obeying the following change rule:

X̄^{i} = X^{j} \frac{∂x̄_{i}}{∂x_{j}}  for X = X̄^{i} \frac{∂}{∂x̄_{i}} .  (1.7)

A 1-form θ on M is a function assigning to each point p ∈ M a cotangent vector θ_{p} ∈ T^{∗}M_{p}. Of course, θ may be viewed as a section of T^{∗}M. In the local coordinates, θ can be expressed by

θ = θ_{i}(x)dx_{i}, θ_{i} = θ(\frac{∂}{∂x_{i}}),

where θ_{i} are obedient to

θ̄_{i} = θ_{j} \frac{∂x_{j}}{∂x̄_{i}}  for θ = θ̄_{i}dx̄_{i}.  (1.8)

In a similar way, we can define (r,s)-tensor fields. For example, let

Φ = Φ_{j_{1}j_{2}}^{i_{1}i_{2}}(x)\frac{∂}{∂x_{i_{1}}} ⊗ \frac{∂}{∂x_{i_{2}}} ⊗ dx_{j_{1}} ⊗ dx_{j_{2}}

be a (2,2)-tensor field. Then we have

Φ̄_{j_{1}j_{2}}^{i_{1}i_{2}} = Φ_{l_{1}l_{2}}^{k_{1}k_{2}} \frac{∂x̄_{i_{1}}}{∂x_{k_{1}}} \frac{∂x̄_{i_{2}}}{∂x_{k_{2}}} \frac{∂x_{l_{1}}}{∂x̄_{j_{1}}} \frac{∂x_{l_{2}}}{∂x̄_{j_{2}}} .

Let φ : M → N be a differentiable map. Its pull-back φ^{∗} maps the function f on N into the function φ^{∗}f on M defined by φ^{∗}f(p) := f(φ(p)) for any p ∈ M. So, the differential dφ ≡ φ_{∗} of φ maps a vector field X on M into a vector field φ_{∗}X on N defined by

φ_{∗}X(f) := X(φ^{∗}f)  (1.9)

for any function f : N → R. Thus, a 1-form ω on N can be pulled back to M written as φ^{∗}ω, which is defined by

φ^{∗}ω(X) = ω(φ_{∗}X)  (1.10)

for any vector field X on M.

4. Riemannian metric. A Riemannian metric (or Riemannian structure) on a differentiable manifold M is a symmetric, positive-definite, (0, 2)−tensor field g on M, i.e., for any vector field X, Y ∈ C(T M),

g(X, Y) = g(Y, X), g(X, X) ≥ 0, (1.11)

where the equality holds if and only if X = 0. In the local coordinates, g can be expressed by

g = g_{ij}(x)dx_{i} ⊗ dx_{j}, g_{ij} = g(\frac{∂}{∂x_{i}} , \frac{∂}{∂x_{j}}),

where g_{ij} = g_{ji} and the matrix (g_{ij}) is positively definite everywhere. The pair (M, g) is called a Riemannian manifold.

Proposition 1.1. There exists a Riemannian metric on a differential manifold M with countable basis.

From (1.11) we see that a Riemannian metric g associates to each point p ∈ M a Euclidean inner product g|_{p} on T M_{p}. We frequently use ⟨X, Y ⟩_{g} (or more simply ⟨X, Y ⟩) instead of g(X, Y). Thus, by means of the Riemannian metric, we can define the norm of a vector and the angle of two vectors as follows:

||X|| := ⟨X, X⟩^{1/2}, cosα = \frac{⟨X, Y ⟩}{||X||||Y ||}.

The simplest example of the Riemannian metric is (1.1). More general, let φ : M → N be an immersion, i.e., dφ is injective everywhere. If h is a Riemannian metric on N, then φ induces a Riemannian metric g on M by defining

g(X, Y) := h(dφ(X), dφ(Y))  (1.12)

for any X, Y ∈ C(T M). Thus, φ : (M, g) → (N, h) is called a isometric immersion. In particular, when M = N and φ is a diffeomorphism satisfying (1.12) with h = g, then φ is called a isometry.

5. Metrics on Lie groups. A Lie group is a group G with a differentiable structure such that the map G × G → G given by (a, b) ↦ ab^{−1}, a, b ∈ G, is differentiable. It follows that the left translation L_{a} and the right translation R_{a} given by L_{a}(b) = ab and R_{a}(b) = ba are diffeomorphiams.

We say a vector field X on G is left invariant if dL_{a}X = X for all a ∈ G. The left invariant vector fields are determined completely by their values at a single point of G. This allows us to introduce an additional structure on the tangent space to the neutral element e ∈ Gin the following manner. To each vector X_{e} ∈ T G_{e} we associate the left invariant field X defined by X_{a} = dL_{a}X_{e} for all a ∈ G. Let X, Y be left invariant vector fields on G. Since for each a ∈ G and for any function f on G,

\begin{aligned}dL_{a}[X, Y]f & = [X, Y](f ∘ L_{a}) = X(dL_{a}Y)f − Y (dL_{a}X)f \\ & = (XY − Y X)f = [X, Y]f, \end{aligned}

we conclude that the bracket of two left invariant vector fields is also left invariant. Thus, we may put [X_{e}, Y_{e}] = [X, Y]_{e} for X_{e}, Y_{e} ∈ T G_{e}. With this operation, T G_{e} becomes a Lie algebra denoted by 𝔤 and called the Lie algebra of G.

A Riemannian metric on G is left invariant if

⟨u, v⟩_{b} = ⟨(dL_{a})_{b}u, (dL_{a})_{b}v⟩_{L_{a}(b)}

for all a, b ∈ G and u, v ∈ T G_{b}, i.e., if L_{a} is an isometry. Analogously, we can define a right invariant Riemannian metric. A Riemannian metric on G which both left and right invariant is called to be bi-invariant.

In order to construct a left invariant metric on G, take an arbitrary inner product ⟨ , ⟩_{e} on 𝔤 and define

⟨u, v⟩_{a} = ⟨(dL_{a^{−1}})_{a}(u), (dL_{a^{−1}})_{a}(v)⟩_{e},  ∀a ∈ G, u, v ∈ T G_{a}.  (1.13)

Clearly, this is a left invariant metric. In a similar manner, we can construct a right invariant metric on G. If G is compact, we may obtain a bi-invariant metric.^{[CE]}

For any a ∈ G, let R_{a^{−1}}L_{a} : G → G be an inner automorphism of G determined by a, which keeps e fixed. Thus, the differential d(R_{a^{−1}}L_{a}) = Ad(a) : 𝔤 → 𝔤 is a linear map (in fact, a homomorphism of 𝔤). Explicitly,

Ad(a)X = dR_{a^{−1}}dL_{a}X = dR_{a^{−1}}X, ∀X ∈ 𝔤.

Let ϕ_{t} be the flow of X ∈ 𝔤. Then we have

[X, Y] = lim_{t→0} \frac{1}{t}\{Y − dϕ_{t}(Y)\}.

Since X is left invariant, L_{b} ∘ ϕ_{t} = ϕ_{t} ∘ L_{b}, giving

ϕ_{t}(b) = ϕ_{t}(L_{b}(e)) = L_{b}(ϕ_{t}(e)) = bϕ_{t}(e) = R_{ϕ_{t}(e)}(b).

Hence, dϕ_{t} = dR_{ϕ_{t}(e)}, and

[X, Y] = lim_{t→0} \frac{1}{t}\{Y − dR_{ϕ_{t}(e)}(Y)\} = lim_{t→0} \frac{1}{t}\{Y − Ad(ϕ_{t}^{−1}(e))Y \}.

Now assume that ⟨ , ⟩ is a bi-invariant metric on G. Then for any X, U, V ∈ 𝔤,

⟨U, V ⟩ = ⟨dR_{ϕ_{t}(e)} ∘ dL_{ϕ_{t}^{−1}(e)}U, dR_{ϕ_{t}(e)} ∘ dL_{ϕ_{t}^{−1}(e)}V ⟩
= ⟨dR_{ϕ_{t}(e)}U, dR_{ϕ_{t}(e)}V ⟩.

Differentiating the above expression with respect to t and setting t = 0, we obtain

⟨[U, X], V ⟩ + ⟨U, [V, X]⟩ = 0.  (1.14)

Conversely, it may be proved that if a positive bilinear form ⟨ , ⟩_{e} defined on 𝔤 satisfies (1.14), then the Riemannian metric defined on G by (1.13) is bi-invariant.

6. Riemannian volume measure. Let (M, g) be a Riemannian manifold. Denote by C(T M) and C(T^{∗}M) the sets of all vector fields and of all 1-forms, respectively. By means of g, we can establish a bijection between C(T M) and C(T^{∗}M) as follows. For X ∈ C(T M), we has uniquely a corresponding 1-form θ defined by

θ(Y) = g(X, Y),  ∀Y ∈ C(T M).  (1.15)

Conversely, given a 1-form θ, (1.15) determined uniquely the corresponding vector field X. In the local coordinates, we see easily that

θ_{i} = g_{ij}X^{i}  and  X^{i} = g^{ij}θ_{i},  (1.16)

where (g^{ij}) = (g_{ij})^{−1}. (r, 0)−tensors are called to be contravariant and (0, r)− tensors are called to be covariant. Thus, by using this bijection, we need not distinguish between the contravariant tensor fields and the covariant tensor fields. An alternating (0, r)−tensor field is called an exterior differential form of degree r, simply a r-form. They play an important rôle in differential geometry.

Let (U, ϕ) be a coordinate neighborhood about a point p ∈ M. Consider a positive orthonormal basis \{e_{i}\} of T M_{p} and write ∂_{i}|_{p} ≡ \frac{∂}{∂x_{i}} |_{p} in the basis \{e_{i}\}: ∂_{i}|_{p} = a_{i}^{j}e_{j}. Then

g_{ij}|_{p} = g(∂_{i}, ∂_{j})|_{p} = a_{i}^{k}a_{j}^{l}g(e_{k}, e_{l}) = ∑_{k} a_{i}^{k}a_{j}^{k}.

By viewing T M_{p} as R^{n}, we see that the volume Vol(∂_{1}, ⋯, ∂_{n})|_{p} of the parallelepiped formed by vectors {∂_{1}|_{p}, ⋯, ∂_{n}|_{p}} in T M_{p} is equal to Vol(e_{1}, ⋯ e_{n}) = 1 multiplied by the determinant of (a_{j}^{i}), so that

Vol(∂_{1}, ⋯, ∂_{n})|_{p} = det (a_{j}^{i}) = \sqrt{det(g_{ij})}|_{p}.

If (V, ψ) is another coordinate neighborhood about p with ḡ_{ij} = g(¯∂_{i}, ¯∂_{j}), ¯∂_{i} ≡ \frac{∂}{∂x̄_{i}}, we have

\sqrt{det(g_{ij})}|_{p} = Vol(∂_{1}, ⋯, ∂_{n})|_{p}
= JVol(¯∂_{1}, ⋯, ¯∂_{n})|_{p} = J\sqrt{det(ḡ_{ij})}|_{p},  (1.17)

where J = det (\frac{∂x_{i}}{∂x̄_{j}} ) > 0 if M is oriented. (1.17) shows that the n−form on M defined by

η = \sqrt{ĝ}dx_{1} ∧ ⋯ ∧ dx_{n},  ĝ := det(g_{ij}).  (1.18)

is well-defined on M, where ∧ denotes the exterior multiplication. The n−form η is called the volume form of (M, g) when M is oriented.

Chapter 2. Connections and Curvatures

1. The Affine connection. Let M be an n−dimensional differentiable manifold and C(T M) be the set of all vector fields on M. An affine connection ∇ on the tangent bundle of M is a map

∇ : C(T M) × C(T M) → C(T M)

dnoted by

(X, Y) ↦ ∇_{X}Y,

which satisfies the following properties:

(i) ∇_{fX+hY} Z = f∇_{X}Z + h∇_{Y} Z; (ii) ∇_{X}(fY + hZ) = (Xf)Y + f∇_{x}Y + (Xh)Z + h∇_{X}Z,

where X, Y, Z ∈ C(T M), and f, h ∈ C(M), the set of all differentiable functions on M. ∇_{X}Y is called the covariant derivative of Y with respect to X.

Let \{x_{i}\} be the local coordinates in an open set U ⊂ M. Write ∂_{i} = \frac{∂}{∂x_{i}}. On putting X = X^{i}∂_{i}, Y = Y^{i}∂_{i} and

∇_{∂_{i}}∂_{j} := Γ_{ij}^{k}∂_{k},  Γ_{ik}^{j}dx_{k} := ω_{i}^{j},  (2.1)

we have

∇_{X}Y = X^{i}(∂_{i}Y^{j} + Γ_{ik}^{j}Y^{k})∂_{j} = {X(Y^{j}) + X^{i}Γ_{ik}^{j}Y^{k}}∂_{j}
= {X(Y^{j}) + X^{i}ω_{i}^{j}(Y)}∂_{j}.  (2.2)

Γ_{ij}^{k} and ω_{i}^{j} in (2.1) are called the connection coefficients and the connection 1-forms of ∇, respectively. They determined completely the connection ∇. It may be shown that there exists an affine connection on a differentiable manifold with countable basis.

The torsion of ∇ is defined by

T (X, Y) = ∇_{X}Y − ∇_{Y} X − [X, Y].  (2.3)

A connection ∇ is said to be torsion free or symmetric if T = 0. From (2.1) and (2.2) it follows that a connection ∇ is symmetric if and only if Γ_{ij}^{k} = Γ_{ji}^{k} in a coordinate neighborhood of M.

Let γ : (a, b) → M be a smooth curve and γ′(t) be the tangent vector field of γ. Let V (t) be a vector field along γ(t). We say that V is parallel along γ if ∇_{γ′}V (t) = 0 for all t. In local coordinates, if

γ : x_{i} = x_{i}(t),  γ′(t) = \frac{dx_{i}}{dt} ∂_{i},  V (t) = V^{i}(t)∂_{i},

then the parallelness of V along γ can be expressed by

\frac{dV^{k}}{dt} + Γ_{ij}^{k} \frac{dx_{i}}{dt} V^{j} = 0, 1 ≤ k ≤ n.

By the theory of ODE, we have from the above

Proposition 2.1. Let M be a differentiable manifold with an affine connection ∇. Let γ : (a, b) → M be a smooth curve and V_{0} ∈ T M_{γ(t_{0})}, t_{0} ∈ (a, b), be a vector. Then there exists a unique parallel vector field V along γ with V (t_{0}) = V_{0}.

V (t) in Proposition 2.1 is called the parallel transport of V_{0} along γ. If γ′ is parallel along γ, the γ is called a geodesic on M with respect to ∇. Clearly, the system of equations for a geodesic on M can be expressed by

\frac{d^{2}x_{i}}{dt^{2}} + Γ_{jk}^{i} \frac{dx_{j}}{dt} \frac{dx_{k}}{dt} = 0.  (2.4)

2. Riemannian connection. Let (M, g) be a Riemannian manifold and ∇ be an affine connection on M. We say that ∇ is compatible with the metric g if

X⟨Y, Z⟩ = ⟨∇_{X}Y, Z⟩ + ⟨Y, ∇_{X}Z⟩  (2.5)

for any X, Y, Z ∈ C(T M), where ⟨ , ⟩ stands for the inner product by g. A Riemannian connection (or Levi-Civita connection) on (M, g) is an affine connection ∇ which is both symmetric and compatible with g.

Theorem (Fundamental theorem of Riemannian geometry). On a Riemannian manifold (M, g) there exists a unique Riemannian connection ∇.

Proof. The symmetric property of ∇ implies that

∇_{X}Y − ∇_{Y} X = [X, Y]  (2.6)

for all X, Y, Z ∈ C(T M). From (2.5) and (2.6) we see easily that

⟨∇_{X}Y, Z⟩ = \frac{1}{2}\{X⟨Y, Z⟩ + Y ⟨Z, X⟩ − Z⟨X, Y ⟩ + ⟨[X, Y], Z⟩

−⟨[Y, Z], X⟩ + ⟨[Z, X], Y ⟩},  (2.7)

which shows that ∇ is unique. Moreover, ∇ defined by (2.7) is an affine connection. □

In fact, using the local coordinates and letting g_{ij} = g(∂_{i}, ∂_{j}), we have from (2.7)

Γ_{ij}^{l}g_{lk} = \frac{1}{2}\{∂_{i}g_{jk} + ∂_{j}g_{ik} − ∂_{k}g_{ij}\},

equivalently,

Γ_{ij}^{l} = \frac{1}{2}g^{lk}\{∂_{i}g_{jk} + ∂_{j}g_{ik} − ∂_{k}g_{ij}\}.  (2.8)

The connection coefficients Γ_{ij}^{l} for the Riemannian connection ∇ are called the Christoffel symbols which are determined uniquely by (2.8).

We now generalize the covariant derivative ∇_{X}Y to tensor fields. Let Φ be a tensor field of order r on (M, g). The covariant derivative of Φ with respect to X is a tensor field ∇_{X}Φ of order r defined by

(∇_{X}Φ)(Y_{1}, ⋯, Y_{r}) = X(Φ(Y_{1}, ⋯, Y_{r})) − ∑_{a=1}^{r} Φ(Y_{1}, ⋯, ∇_{X}Y_{a}, ⋯, Y_{r}), (2.9)

where X, Y_{1}, ⋯, Y_{r} ∈ C(T M). Clearly, for a function f on M, we have

∇_{X}f = Xf = df(X).

The covariant differential of Φ is a tensor field ∇Φ of order (r + 1) defined by

∇Φ(Y_{1}, ⋯ Y_{r}, X) = (∇_{X}Φ)(Y_{1}, ⋯ Y_{r}).  (2.10)

If ∇Φ = 0, then Φ is said to be parallel with respect to ∇. From (2.9) it follows that

(∇_{X}g)(Y, Z) = X⟨Y, Z⟩ − ⟨∇_{X}Y, Z⟩ − ⟨Y, ∇_{X}Z⟩.

This implies the following

Proposition 2.1. A symmetric affine connection ∇ on a Riemannian manifold (M, g) is Riemannian if and only if g is parallel with respect to ∇.

3. Curvature tensors. The curvature R of the Riemannian connection ∇ on a Riemannian manifold (M, g) is an operator which associates to each pair X, Y ∈ C(T M) a map R(X, Y) : C(T M) → C(T M) defined by

R(X, Y)Z = ∇_{X}∇_{Y} Z − ∇_{Y} ∇_{X}Z − ∇_{[X,Y]}Z  (2.11)

for any Z ∈ C(T M). It is easy to check that the triplet R(X, Y)Z called the curvature tensor is function-linear with respect to every entry. By using the Jacobi identity for vector fields, the curvature tensor R(X, Y)Z satisfies the following (first) Bianchi identity:

R(X, Y)Z + R(Y, Z)X + R(Z, X)Y = 0.  (2.12)

Set

R(X, Y, Z, W) := ⟨R(Z, W)Y, X⟩,  (2.13)

which is a (0,4)-tensor field and called also the curvature tensor. By (2.11) and (2.13), one can see easily that

(i) R(X, Y, Z, W) = −R(Y, X, Z, W) = −R(X, Y, W, Z).

(ii) R(X, Y, Z, W) = R(Z, W, X, Y).

By (2.9) and (2.10), the curvature tensor (2.13) satisfies the following second Bianchi identity:

∇R(V, W, X, Y, Z) + ∇R(V, W, Y, Z, X) + ∇R(V, W, Z, X, Y) = 0.  (2.14)

In the local coordinates, write

R(∂_{k}, ∂_{l})∂_{j} = R_{jkl}^{i}∂_{i}, R(∂_{i}, ∂_{j}, ∂_{k}, ∂_{l}) = R_{ijkl} = g_{im}R_{jkl}^{m}.

We have

R_{jkl}^{i} = ∂_{k}Γ_{jl}^{i} − ∂_{l}Γ_{jk}^{i} + Γ_{jl}^{m}Γ_{mk}^{i} − Γ_{jk}^{m}Γ_{ml}^{i},

R_{ijkl} = −R_{jikl} = −R_{ijlk} = R_{klij},
R_{jkl}^{i} + R_{klj}^{i} + R_{ljk}^{i} = 0.

Moreover, on putting

R_{ijkl,h} := ∂_{h}R_{ijkl} − R_{mjkl}Γ_{ih}^{m} − R_{imkl}Γ_{jh}^{m} − R_{ijml}Γ_{kh}^{m} − R_{ijkm}Γ_{lh}^{m}, (2.14) can be written as

R_{ijkl,h} + R_{ijlh,k} + R_{ijhk,l} = 0.

Note that the condition that ∇g = 0 is equivalent to that

g_{ij,k} := ∂_{k}g_{ij} − g_{lj}Γ_{ik}^{l} − g_{il}Γ_{jk}^{l} = 0.

By (2.10), we have the second differential defined by

∇^{2}Φ(⋯, X, Y) = (∇_{Y} (∇_{X}Φ))(⋯) − ∇_{∇_{Y} X}Φ(⋯).

On putting

R(X, Y)Φ := ∇_{X}∇_{Y} Φ − ∇_{Y} ∇_{X}Φ − ∇_{[X,Y]}Φ,

we have the following Ricci formula:

∇^{2}Φ(⋯, X, Y) − ∇^{2}Φ(⋯, Y, X) = −R(X, Y)Φ(⋯).  (2.14’)

4. Curvatures. Let (M, g) be a Riemannian manifold and p ∈ M. Let σ ⊂ T M_{p} be a 2-dimensional subspace, and X, Y ∈ σ be two linearly independent vectors. The sectional curvature of σ at p is defined by

\begin{matrix}K(σ) = K(X ∧ Y) := \frac{R(X, Y, X, Y)}{S(X, Y, X, Y)}, \\ S(X, Y, Z, W) := ⟨X, Z⟩⟨Y, W ⟩ − ⟨X, W ⟩⟨Y, Z⟩. \end{matrix}  (2.15)

It is easy to check that K(σ) does not depend on the choice of the vectors X, Y ∈ σ. The importance of the sectional curvature comes from the fact that knowledge of K(σ), for all σ, determines the curvature tensor R(X, Y, Z, W) completely. In particular, the sectional curvature is just the Gauss curvature when dimM = 2. In the local coordinates {x_{i}}, (2.15) can be expressed by

K(X ∧ Y) = \frac{R_{ijkl}X^{i}Y^{j}X^{k}Y^{l}}{(g_{ik}g_{kl} − g_{il}g_{jk})X^{i}Y^{j}X^{k}Y^{l}}

for X = X^{i}∂_{i} and Y = Y^{i}∂_{i}.

A (M, g) is said to have constant (sectional) curvature if K(σ) = c (=const.) for all p ∈ M and all σ ∈ T M_{p}. It is obvious from (2.15) that (M, g) has constant curvature c if and only if

R(X, Y, X, Y) = cS(X, Y, X, Y), ∀X, Y ∈ C(T M).

This condition can be weakened as follows.

Theorem (Schur’s theorem). If dimM = n ≥ 3 and K(σ)(p) = λ(p) at any p ∈ M for any σ ⊂ T M_{p}, then λ(p) is constant, namely, (M, g) has constant curvature.

It can be proved by applying the second Bianchi identity.

Let X ∈ T M_{p} be a unit vector. We take an orthonormal basis {e_{1} = X, e_{2}, ⋯ e_{n}} in T M_{p} and consider the following ”averages”:

Ric_{p}(X) := ∑_{j=2}^{n} K(X ∧ e_{j})_{p},  (2.16)
ϱ(p) := ∑_{i} Ric_{p}(e_{i}),  (2.17)

which are called the Ricci curvature in the direction X and the scalar curvature at p, respectively. It is easy to prove that the expressions (2.16) and (2.17) do not depend on the choice of the corresponding orthonormal basis.

In order to explain these facts, we define a bilinear form r on T M_{p} as follows: let X, Y ∈ T M_{p} and put

r(X, Y) = trace of the map :Z ↦ R(Z, X)Y,  (2.18)

which is called the Ricci tensor. In the above orthonormal basis {e_{1} = X, e_{2}, ⋯, e_{n}}, we have

r(X, Y) = ∑_{j}⟨R(e_{j}, X)Y, e_{j}⟩ = ∑_{j}⟨R(e_{j}, Y)X, e_{j}⟩ = r(Y, X),

i.e., r is symmetric and

r(X, X) = ∑_{j} R(X, e_{j}, X, e_{j}) = ∑_{j} K(X ∧ e_{j}) = Ric(X).

On the other hand, the form r on T M_{p} corresponds to a linear self-adjoint map L_{r} given by ⟨L_{r}(X), Y ⟩ = r(X, Y). In an orthonormal basis {e_{1}, ⋯, e_{n}}, we have from (2.17)

ϱ = ∑_{i} Ric(e_{i}) = ∑_{i} r(e_{i}, e_{i}) = ∑_{i}⟨L_{r}(e_{i}), e_{i}⟩ = trace of L_{r}.

In the local coordinates {x_{i}}, these can be expressed as

r(X, Y) = R_{ij}X^{i}Y^{j} with R_{ij} = g^{kl}R_{ikjl},
ϱ = g^{ik}R_{ik} = g^{ik}g^{jl}R_{ijkl}.

A Riemannian manifold (M, g) is called an Einstein manifold if there is a constant λ on M such that r = λg. In such a case, g is called the Einstein metric, and the scalar curvature ϱ = nλ is constant. By using the second Bianchi identity, we can easily to prove the following

Proposition 2.2. If dimM = n ≥ 3 and r(p) = λ(p)g(p) for all point p ∈ M, then λ must be independent of p, i.e., λ is constant. Every 2 or 3-dimensional Einstein manifold must be of constant curvature.

Let G be a Lie group with a bi-invariant Riemannian metric ⟨ , ⟩. Let X, Y, Z, W ∈ 𝔤 be the left invariant vector fields on G. Then we have (cf. [CE], Ch. 3)

(1) ∇_{X}Y = \frac{1}{2}[X, Y];

(2) ⟨R(X, Y)Z, W ⟩ = \frac{1}{4}(⟨[X, W], [Y, Z]⟩ − ⟨[X, Z], [Y, W]⟩);

(3) R(X, Y, X, Y) = \frac{1}{4}||[X, Y]||^{2}. Hence, the sectional curvature is nonnegative.

5. Conformal change. Let M be an n−dimensional differentiable manifold. Two Riemannian metrics g and ḡ on M are said to be conformal if there exists a positive function φ : M → R such that

ḡ = φ^{2}g.  (2.19)

In the local coordinates {x_{i}}, (2.19) is reduced to

ḡ_{ij} = φ^{2}g_{ij},  (2.19’)

which implies that the angle between two vectors is preserved. Let ¯∇ and ∇ be the Riemannian connections corresponding to ḡ and g. Then the coefficients Γ̄_{jk}^{i} and Γ_{jk}^{i} of ¯∇ and ∇ satisfy that

Γ̄_{jk}^{i} = Γ_{jk}^{i} + δ_{j}^{i}φ_{k} + δ_{k}^{i}φ_{j} − φ^{i}g_{jk},  (2.20)

where

φ_{j} = ∂_{j} log φ,  φ^{i} = g^{ij}φ_{j}.  (2.21)

On putting

φ_{ij} := φ_{i,j} − φ_{i}φ_{j} + \frac{1}{2}φ_{k}φ^{k}g_{ij},  φ_{i,j} = ∂_{j}φ_{i} − φ_{k}Γ_{ij}^{k},  (2.22)

a straightforward calculation gives that

φ^{−2}R̄_{ijkl} = R_{ijkl} + g_{il}φ_{jk} − g_{ik}φ_{jl} + g_{jk}φ_{il} − g_{jl}φ_{ik},  (2.23)
R̄_{ij} = R_{ij} − (n − 2)φ_{ij} − g^{kl}φ_{kl}g_{ij}.  (2.24)

Hence, the scalar curvatures ϱ̄ and ϱ satisfy that

φ^{2}ϱ̄ = ϱ − 2(n − 1)g^{ij}φ_{ij}.  (2.25)

If we set

L_{ij} := \frac{1}{n − 2}\{R_{ij} − \frac{ϱ}{2(n − 1)}g_{ij}\}  (2.26)

called the Schouten tensor, then

L̄_{ij} = L_{ij} − φ_{ij}.  (2.27)

From (2.23) ∼ (2.27) it follows that the following tensor

W_{ijkl} := R_{ijkl} + g_{il}L_{jk} − g_{ik}L_{jl} + L_{il}g_{jk} − L_{ik}g_{jl}

satisfies that

φ^{−2}W̄_{ijkl} = W_{ijkl},

which implies that

W̄_{jkl}^{i} = W_{jkl}^{i}  (2.28)

called the Weyl conformal curvature tensor. The following proposition is well known.^{[BS]}

Proposition 2.3. For n = 3, the Weyl tensor W vanishes identically. For n ≥ 4, W vanishes identically if and only if g is (locally) conformally flat, i.e., there exists φ > 0 such that the metric (2.19) is flat, namely, R̄ ≡ 0.

In 1960 H. Yamabe proposed the following proposition: In an n(≥ 3)-dimensional compact Riemannian manifold (M, g) there exists a smooth positive function φ such that the conformal metric ḡ = φ^{2}g has constant scalar curvature. It is equivalent to solving the equation (2.25) with ϱ̄ =const. on (M, g). This problem so-called Yamabe’s conjecture was solved finally by R. Schoen (see J. Diff. Geom., 20(1984), 479-495).

Chapter 3. Harmonic Forms and Hodge Theory

1. Exterior differential. Let M be an n−dimensional differentiable manifold. Recall that an r-form θ on M is an alternating (0,r)-tensor field (Lec.1, §6). In the local coordinates {x_{i}}, θ can be expressed by

θ = θ_{i_{1}···i_{r}}(x)dx^{i_{1}} ∧ ⋯ ∧ dx^{i_{r}}.  (3.1)

Denote by Λ^{r}(M) the set of all r−forms on M. Clearly, Λ^{0}(M) = C(M) is the set of all differential function on M. By the rule of the exterior multiplication, we see that 0 ≤ r ≤ n.

The exterior differential operator is a map d : Λ^{r}(M) → Λ^{r+1}(M), which satisfies the following conditions:

(i) d is R−linear; (ii) For f ∈ Λ^{0}(M), df is the usual differential of f, and d(df) = 0; (iii) d(θ ∧ σ) = dθ ∧ σ + (−1)^{r} θ ∧ dσ for θ ∈ Λ^{r}(M) and any form σ.

We can claim the exterior differential operator d is of existence and uniqueness. In fact, d is a local operator. We need only to define it in each coordinate neighborhood of M. For any θ ∈ Λ^{r}(M) denoted by (3.1), we define

\begin{aligned}dθ & = dθ_{i_{1}···i_{r}} ∧ dx_{i_{1}} ∧ ⋯ ∧ dx_{i_{r}} \\ & = (∂_{j}θ_{i_{1}···i_{r}})dx_{j} ∧ dx_{i_{1}} ∧ ⋯ ∧ dx_{i_{r}} ∈ Λ^{r+1}(M). \end{aligned}  (3.2)

It is easy to check that the above conditions are satisfied. Moreover, we have

Lemma 3.1(The Poincaré lemma).

d^{2} = 0,

i.e., d(dθ) = 0 for any θ ∈ Λ^{r}(M).

A simple computation gives

Lemma 3.2. For ω ∈ Λ^{1}(M) and X, Y ∈ C(T M),

dω(X, Y) = X(ω(Y)) − Y (ω(X)) − ω([X, Y]).

Let M and N be differentiable manifolds, and φ : M → N be a differentiable map. Denote by d_{M} and d_{N} the exterior differential operators on M and N, respectively. It can be proved that for any ψ ∈ Λ^{r}(N) we have^{[BS]}

φ^{∗}(d_{N}ψ) = d_{M}(φ^{∗}ψ).

Simply, we write

φ^{∗} ∘ d = d ∘ φ^{∗}.  (3.3)

Now we assume that M is compact. A θ ∈ Λ^{r}(M) is said to be closed if dθ = 0. Moreover, θ is said to be exact if there is a σ ∈ Λ^{r−1}(M) such that dσ = θ. By Lemma 3.1, an exact form is closed. Let Z^{r}(M, R) denote the set of all closed r−forms on M; since Z^{r}(M) is the kernel of the homomorphism d : Λ^{r}(M) → Λ^{r+1}(M) it is a linear subspace of Λ^{r}(M). Similarly the set B^{r}(M, R) of all exact r−forms on M is the image of d : Λ^{r−1}(M) → Λ^{r}(M) and thus a linear subspace. Moreover, by Poincaré lemma, d^{2} = 0 implies that B^{r}(M, R) ⊂ Z^{r}(M, R) which allows us to form the quotient space H^{r}(M, R) := Z^{r}(M, R)/B^{r}(M, R) called the de Rham cohomology group of dimension r. Set

H^{∗}(M, R) = H^{0}(M, R) ⊕ H^{1}(M, R) ⊕ ⋯ ⊕ H^{n}(M, R),

which is an algebra with the exterior multiplication. The great importance of the de Rham cohomology groups and the de Rham algebra H^{∗}(M, R) stems from de Rham theorem:

Theorem (the de Rham theorem). There is a natural isomorphism of H^{∗}( M, R) and the cohomology ring of M under which H^{r}(M, R) corresponds to the r-th cohomology group of M.

In fact, let S_{∞}^{r}(M, R) denote the set of all differentiable singular r-cochain on M, of which elements assign to each differentiable singular r-simplex in M a real number of R. We can define homomorphisms

k_{r} : Λ^{r}(M) → S_{∞}^{r}(M, R)

for each integer r ≥ 0 by setting

k_{r}(ω)(σ) := ∫_{σ} ω

for each differentiable r-form ω on M and differentiable singular r-simplex σ in M. It is an immediate consequence of Stokes’ theorem that the homomorphisms k_{r} induce a cochain map

k : Λ^{∗}(M) → S_{∞}^{∗}(M, R).

Let

k_{r}^{∗} : H^{r}(M, R) → Ȟ^{r}(M, R)

denote the induced homomorphism of the cohomology modules (real vector spaces), where Ȟ^{r}(M, R) stands for the r-th singular cohomology group which isomorphic to the r-th cohomology group of M. k_{r}^{∗} is called the de Rham homomorphism. The de Rham theorem says that k_{r}^{∗} is the canonical isomorphism for each integer r. For the detail, one may be referred to [Wa].

2. Moving frames. Let (M, g) be an n−dimensional Riemannian manifold and ∇ be the Riemannian connection. Since ∇ is a symmetric affine connection, we see easily that for any θ ∈ Λ^{r}(M)

(dθ)(X_{1}, ⋯, X_{r+1}) = ∑_{i=1}^{r+1}(−1)^{i+1}(∇_{X_{i}}θ)(X_{1}, ⋯, X̂_{i}, ⋯, X_{r+1}),  (3.4)

or equivalently, in the local coordinates and by (3.2),

dθ = θ_{i_{1}···i_{r},j}dx_{j} ∧ dx_{i_{1}} ∧ ⋯ ∧ dx_{i_{r}},

where θ_{i_{1}···i_{r},j} are defined in the similar way to R_{ijkl,h} in Ch.2, §3.

Let {e_{i}} be a local field of orthonormal frames in an open set U ⊂ M, and {ω^{i}} be the dual field of coframes. Since g(e_{i}, e_{j}) = δ_{ij} we have

g = ∑_{i} ω^{i} ⊗ ω^{i}.

The connection 1-forms can be written by

∇_{X}e_{j} := ω_{j}^{i}(X)e_{i} ∀X ∈ C(T M).

Noting that X = ω^{i}(X)e_{i}, the symmetricity of ∇ implies that

\begin{aligned}0 & = ∇_{X}(ω^{i}(Y)e_{i}) − ∇_{Y} (ω^{i}(X)e_{i}) − ω^{i}([X, Y])e_{i} \\ & = \{X(ω^{i}(Y)) − Y (ω^{i}(X)) − ω^{i}([X, Y])\}e_{i} + \{ω^{j}(Y)ω_{j}^{i}(X) − ω^{j}(X)ω_{j}^{i}(Y)\}e_{i}, \end{aligned}

from which and Lemma 3.2 it follows that

dω^{i} = −ω_{j}^{i} ∧ ω^{j}.  (3.5)

Moreover, since ∇ is compatible with g we have

ω_{j}^{i} + ω_{i}^{j} = 0.  (3.6)

Similarly, if we define 2-forms Ω_{j}^{i} by

R(X, Y)e_{i} := Ω_{i}^{j}(X, Y)e_{j},

then, by the definition of R(X, Y) and Lemma 3.2 we have

dω_{j}^{i} = −ω_{k}^{i} ∧ ω_{j}^{k} + Ω_{j}^{i}.  (3.7)

These are called the structure equations of (M, g). Putting

Ω_{j}^{i} = \frac{1}{2}R_{jkl}^{i}ω^{k} ∧ ω^{l}  (3.8)

and taking exterior differential in both sides of (3.5) and (3.7), we can obtain the Bianchi identities:

R_{jkl}^{i} + R_{klj}^{i} + R_{ljk}^{i} = 0,

R_{jklh}^{i} + R_{jlhk}^{i} + R_{jhkl}^{i} = 0,

where R_{jklh}^{i} are defined by

R_{jklh}^{i}ω^{h} = DR_{jkl}^{i} := dR_{jkl}^{i} − R_{hkl}^{i}ω_{j}^{h} − R_{jhl}^{i}ω_{k}^{h} − R_{jkh}^{i}ω_{l}^{h} + R_{jkl}^{h}ω_{h}^{i},

which are just the covariant differential of R_{jkl}^{i}.

3. Harmonic forms. On (M, g) we can introduce the local inner product in Λ^{r}(M) as follows. For α = α_{i_{1}···i_{r}}dx_{i_{1}} ∧ ⋯ ∧ dx_{i_{r}} and β = β_{i_{1}···i_{r}}dx_{i_{1}} ∧ ⋯ ∧ dx_{i_{r}},

⟨α, β⟩ := α_{i_{1}···i_{r}}β^{i_{1}···i_{r}} = g^{i_{1}j_{1}} ⋯ g^{i_{r}j_{r}}α_{i_{1}···i_{r}}β_{j_{1}···j_{r}},

which is independent of the choice of coordinates.

We now consider the Hodge star operator ∗ to be an isomorphism Λ^{r}(M) → Λ^{n−r}(M) defined by

α ∧ (∗β) = ⟨α, β⟩η  (3.9)

for any α, β ∈ Λ^{r}(M), where η is the volume n-form of (M, g) (cf. Ch.1, §6). It is easy to see that ∗ operator has the following properties:

\begin{matrix}∗ (α + β) = ∗α + ∗β;  ∗(fα) = f(∗α); \\ ∗ ∗α = (−1)^{r(n−r)}α;  α ∧ ∗β = β ∧ ∗α, \end{matrix}  (3.10)

where α, β ∈ Λ^{r}(M) and f ∈ C(M). Hence, ∗1 = η, and α ∧ ∗α = 0 if and only if α = 0.

As the formal adjoint operator of the exterior differential d, the codifferential operator δ : Λ^{r}(M) → Λ^{r−1}(M) is defined by

δ := (−1)^{n(r+1)+1} ∗ d ∗ ∀r ≥ 1,  δf = 0 ∀f ∈ C(M).  (3.11)

It is easy to check that the codifferential δ has the following properties:

δ^{2} = 0;  d ∗ δ = δ ∗ d = 0;  ∗dδ = δd∗,  ∗δd = dδ∗,
∗δα = (−1)^{r}d ∗ α, δ ∗ α = (−1)^{1−r^{2}} ∗ dα, ∀α ∈ Λ^{r}(M).

A form α ∈ Λ^{r}(M), r ≥ 1, satisfying δα = 0 is called co-closed; α is called co-exact if there is a β ∈ Λ^{r+1}(M) such that δβ = α. Clearly, a co-exact form is co-closed.

We now introduce the global inner product ( , ) in Λ^{r}(M). Suppose that M is compact and oriented. For α, β ∈ Λ^{r}(M),

(α, β) := ∫_{M} α ∧ ∗β = ∫_{M}⟨α, β⟩η.  (3.12)

Clearly, (α, β) = (β, α) and (α, α) ≥ 0 where the equality holds if and only if α = 0.

Proposition 3.1. Let M be compact and oriented. For α ∈ Λ^{r}(M) and β ∈ Λ^{r+1}(M), we have

(α, δβ) = (dα, β).

Proof. Since

d(α ∧ ∗β) = dα ∧ ∗β + (−1)^{r}α ∧ d ∗ β,

then, by the Stokes formula, we have

(dα, β) = (−1)^{r+1} ∫_{M} α ∧ d ∗ β = ∫_{M} α ∧ ∗δβ = (α, δβ).

□

In particular, we have

∫_{M}(δα)η = ∫_{M}(δα) ∗ 1 = 0,  ∀α ∈ Λ^{1}(M).  (3.13)

By using Proposition 3.1 and (3.4), we can easily see that for α ∈ Λ^{r}(M),

δα(X_{1}, ⋯ X_{r−1}) = − ∑_{i=1}^{n}(∇_{e_{i}}α)(e_{i}, X_{1}, ⋯ X_{r−1}),  (3.14)

where {e_{i}} is a local field of orthonormal frames on M and X_{1}, ⋯ X_{r−1} ∈ C(T M).

For a X ∈ C(T M), its divergence divX (or δX) is the codifferential of its dual 1-form X^{♭}, i.e., divX = δX^{♭}. By (3.14), we have

divX = − ∑_{i=1}^{n}⟨∇_{e_{i}}X, e_{i}⟩.  (3.14’)

The operator Δ := dδ + δd : Λ^{r}(M) → Λ^{r}(M) is called the Laplacian in Λ^{r}(M). It is easy to check that Δ has the following properties:

Δ = (d + δ)^{2}; d ∘ Δ = Δ ∘ d = −d ∘ δ ∘ d;
δ ∘ Δ = Δ ∘ δ = −δ ∘ d ∘ δ; ∗Δ = Δ ∗ .

A form α ∈ Λ^{r}(M), r ≥ 0, satisfying Δα = 0 is called to be harmonic.

Proposition 3.2. Let (M, g) be a compact oriented Riemannian manifold without boundary. An α ∈ Λ^{r}(M) is harmonic if and only if α is both closed and co-closed.

Proof. By Proposition 3.1, we have

(Δα, α) = (dδα + δdα, α) = (δα, δα) + (dα, dα) ≥ 0,

which implies that Δα = 0 if and only if δα = 0 and dα = 0. □

Hence, Δ is an elliptic self-adjoint operator, i.e.,

(Δα, β) = (α, Δβ), ∀α, β ∈ Λ^{r}(M).

In particular, for f ∈ C(M), Δf = δdf = −tr∇df because of δf = 0. In the local coordinates {x_{i}}, we have

Δf = −g^{ij}∇df(∂_{i}, ∂_{j}) = −\frac{1}{\sqrt{ĝ}} ∑_{i,j} ∂_{i}(\sqrt{ĝ}g^{ij}∂_{j}f),  ĝ = det(g_{ij}).  (3.15)

For α ∈ Λ^{1}(M), let α^{#} be the dual vector field to α. Then it is easy to see that

Δα(X) = -tr∇^{2}α(X) + r(α^{#}, X),  (3.16)

where r is the Ricci tensor of (M, g) and tr∇^{2} := ∑_{i} ∇_{e_{i}}∇_{e_{i}} for an orthonormal basis {e_{i}} on M.

4. Hodge Theory. Let (M, g) be an n-dimensional compact oriented Riemannian manifold without boundary. It is an interesting problem to find necessary and sufficient conditions for there to exist a solution ω of the equation Δω = σ for given σ ∈ Λ^{r}(M). Set

H^{r}(M) := {α ∈ Λ^{r}(M)|Δα = 0} = kerΔ.

Suppose that ω is a solution of Δω = σ. Then for any α ∈ H^{r}(M),

(σ, α) = (Δω, α) = (ω, Δα) = 0,

namely, σ must be orthogonal to H^{r}(M), denoted by σ⊥H^{r}(M). The Hodge theorem shows that it is sufficient. We introduce

\begin{matrix}dΛ^{r−1}(M) := \{dα|α ∈ Λ^{r−1}(M)\} ⊂ Λ^{r}(M), \\ δΛ^{r+1}(M) := \{δα|α ∈ Λ^{r+1}(M)\} ⊂ Λ^{r}(M). \end{matrix}

Theorem (The Hodge decomposition theorem). For each integer 0 ≤ r ≤ n, H^{r}(M) is finite dimensional, and we have the following orthogonal direct sum decompositions of the space Λ^{r}(M):

\begin{aligned}Λ^{r}(M) & = Δ(Λ^{r}(M)) ⊕ H^{r}(M) \\ & = dδΛ^{r}(M) ⊕ δdΛ^{r}(M) ⊕ H^{r}(M) \\ & = d(Λ^{r−1}(M)) ⊕ δ(Λ^{r+1}(M)) ⊕ H^{r}(M). \end{aligned}  (3.17)

Consequently, the equation Δω = σ has a solution ω ∈ Λ^{r}(M) if and only if σ⊥H^{r}(M).

Suppose that ω is a solution of Δω = σ. Then

(ω,Δα) = (Δω, α) = (σ, α)  ∀α ∈ Λ^{r}(M).  (3.18)

Thus, we can view a solution of Δω = σ as a certain type of linear functional on Λ^{r}(M), namely, ω determines a bounded linear functional Φ on Λ^{r}(M) by

Φ(α) = (ω, α);

and in view of (3.18), the functional Φ satisfies

Φ(Δα) = (σ, α)  for all α ∈ Λ^{r}(M).  (3.19)

This view of a solution turns out to be extremely useful, for it will allow us to bring various techniques of functional analysis to bear on the problem of solving Δω = σ. We shall call such a linear functional a weak solution of Δω = σ. That is, a weak solution of Δω = σ is a bounded linear functional Φ : Λ^{r}(M) → R satisfying (3.19).

The proof of the theorem is based on the following two facts:

(1) Regurality. Let α ∈ Λ^{r}(M), and let Φ be a weak solution of Δω = σ. Then there exists ω ∈ Λ^{r}(M) such that Φ(α) = (ω, α) for each α ∈ Λ^{r}(M). Consequently, Δω = σ.

(2) Convergence. Let {α_{j}} be a sequence of smooth r-forms on M such that ||α_{j}|| ≤ c and ||Δα_{j}|| ≤ c for all j and for some constant c > 0. Then a subsequence of {α_{j}} is a Cauchy sequence in Λ^{r}(M).

Now, if H^{r}(M) were not finite dimensional, then H^{r}(M) would contain an infinite orthonormal sequence. But by Convergence (2), this orthonormal sequence would contain a Cauchy subsequence, which is impossible. Hence, H^{r}(M) is finite dimensional.

It is sufficient to prove the decomposition in the first line of (3.17), for the other two lines of (3.17) then follows from the definitions and Proposition 3.1.

Let {ω_{1}, ⋯ ω_{d}} be an orthonormal basis of H^{r}(M). Then an arbitrary form α ∈ H^{r}(M) can uniquely be written as

α = β + ∑_{i=1}^{d}(α, ω_{i})ω_{i},

where β lies in (H^{r}(M))^{⊥}, the subspace of Λ^{r}(M) consisting of all elements orthogonal to H^{r}(M). Thus, we have an orthogonal direct sum decomposition

Λ^{r}(M) = (H^{r}(M))^{⊥} ⊕ H^{r}(M).

It can be prove that (H^{r}(M))^{⊥} = Δ(Λ^{r}(M)). Thus, the theorem is proved. For the detail, one may be referred to [Wa].

The great importance of harmonic forms stems from the following

Theorem 3.1. Each de Rham cohomology class on (M, g) contains a unique harmonic representative.

Proof. Let h : Λ^{r}(M) → H^{r}(M) be the orthogonal projection related to (3.17). If ω ∈ Λ^{r}(M) is closed, then from (3.17) it follows that

ω = dα + h(ω) ∀α ∈ Λ^{r−1}(M),

which implies that (cf. §1.)

[ω] = [h(ω)] ∈ H^{r}(M, R).

Since H^{r}(M)⊥dΛ^{r−1}(M) we see that two different harmonic forms must belong to two different de Rham cohomology classes. In fact, if γ_{1}, γ_{2} ∈ H^{r}(M) and [γ_{1}] = [γ_{2}], then we have

γ_{1} − γ_{2} = dα ∀α ∈ Λ^{r−1}(M).

But, by the Hodge theorem, dα⊥(γ_{1} − γ_{2}). Thus, the above equality implies that

dα = 0, γ_{1} = γ_{2}.

Hence, h(ω) is unique in H^{r}(M, R). □

From Theorem 3.1 it follows that H^{r}(M) is isomorphic to H^{r}(M, R), so that each H^{r}(M, R) is finite dimensional. The r-th Betti number β_{r}(M) of (M, g) is defined by

β_{r}(M) = dimH^{r}(M, R) = dimH^{r}(M), 0 ≤ r ≤ n.

The Euler-Poincaré characteristic number χ(M) of (M, g) is defined by

χ(M) = ∑_{r=0}^{n}(−1)^{r}β_{r}(M).  (3.20)

On the other hand, it can be proved that (cf. [BS])

H^{r}(M) ≅ H^{n−r}(M),

so that β_{r}(M) = β_{n−r}(M). This implies that χ(M) = 0 if n = dimM is odd.

Proposition 3.3. Let (M, g) be a compact oriented Riemannian manifold without boundary. If its Ricci curvature is positive, then

β_{1}(M) = β_{n−1}(M) = 0.

Proof. Let α ∈ H^{1}(M) and α^{#} be the dual field to α. By (3.16) we see that

0 = Δ(α^{#}) = −tr(∇^{2}α)(α^{#}) + r(α^{#}, α^{#}),

from which and (3.13) it follows that

0 = − ∫_{M} tr(∇^{2}α)(α^{#})η + ∫_{M} r(α^{#}, α^{#})η
= ∫_{M} |∇α|^{2}η + ∫_{M} r(α^{#}, α^{#})η ≥ 0.

Hence, r(α^{#}, α^{#}) = 0, which implies that α^{#} = 0, i.e., α = 0 since the Ricci curvature of (M, g) is positive. □

Chapter 4. Geodesics and Completeness

1. Geodesics. Let (M, g) be an n-dimensional Riemannian manifold, ∇ the Riemannian connection. A differentiable curve γ : [a, b] → M on M is geodesic (cf. Ch.1, §1) if its tangent vector field γ′(t) = dγ(∂/∂t), t ∈ [a, b], satisfies

∇_{γ′}γ′ = 0.  (4.1)

Since γ′⟨γ′, γ′⟩ = 2⟨∇_{γ′}γ′, γ′⟩ = 0, the length ||γ′|| = ⟨γ′, γ′⟩^{1/2} of γ′ is invariant along γ. Thus,

s(t) = ∫_{a}^{t} ||γ′||dt = ||γ′||(t − a),

i.e., the arc length s(t) of γ is a linear function of the parameter t. The parameter is actually arc length if and only if ||γ′|| = 1, in which case we say the geodesic γ is normalized.

In the local coordinates \{x_{i}\}, writing γ′(t) = \frac{dx_{i}}{dt} ∂_{i}, (4.1) is reduced to

\frac{d^{2}x_{i}}{dt^{2}} + Γ_{jk}^{i} \frac{dx_{j}}{dt} \frac{dx_{k}}{dt} = 0,  (4.2)

where Γ_{jk}^{i} is the Christoffel symbol. By the theory of ODE, given a point p ∈ M and a vector v ∈ T M_{p} with ||v|| < δ (some positive constant), there exists a unique geodesic γ_{v} : (−ε, ε) → M, t ↦ γ_{v}(t), such that γ_{v}(0) = p and γ′_{v}(0) = v. For any constant a > 0, we define a curve σ : (−ε/a, ε/a) → M by

σ(t) = γ_{v}(at) ∀t ∈ (−ε/a, ε/a).

Clearly, σ is a geodesic on M with σ(0) = p and σ′(0) = av because ∇_{σ′}σ′ = a^{2}∇_{γ′}γ′ = 0. Therefore, σ = γ_{av}, i.e.,

γ_{v}(at) = γ_{av}(t).  (4.3)

(4.3) permits us to make the interval of definition of a geodesic uniformly large in a neighborhood of p ∈ M. So, we can define the exponential map exp_{p} : T M_{p} → M by

exp_{p}(v) = γ_{v}(1) ∀v ∈ T M_{p}.

Geometrically, exp_{p}(v) is a point of M obtained by going out the length equal to ||v||, starting from p, along the geodesic γ which passes through p with velocity equal to v/||v||.

Let us calculate d(exp_{p})_{o}:

d(exp_{p})_{o}(v) = \frac{d}{dt}(exp_{p}(tv))|_{t=0} = \frac{d}{dt}(γ_{tv}(1))|_{t=0} = \frac{d}{dt}(γ_{v}(t))|_{t=0} = v.

Hence, d(exp_{p})_{o} is the identity of T M_{p}, and it follows from the inverse function theorem that exp_{p} is a local diffeomorphism in a neighborhood of 0 on T M_{p}.

Recall that T M is the set of pairs (p, v) where p ∈ M and v ∈ T M_{p}. If {x_{i}} are the local coordinates in a neighborhood U ⊂ M containing p, then any vector v ∈ T M_{p} can be written as y^{i}∂_{i}. Thus, we can take (x_{1}, ⋯, x_{n}, y^{1}, ⋯, y^{n}) as coordinates of (p, v) in T M. In what follows we identify the tangent space T (T M_{p})_{v} to T M_{p} at v ∈ T M_{p} with T M_{p} itself by identifying ∂/∂y^{i} with ∂_{i} = ∂/∂x_{i}.

Lemma 4.1 (Gauss). Let p ∈ M and v ∈ T M_{p} such that exp_{p}v is defined. Let w ∈ T M_{p} ≅ T (T M_{p})_{v}. Then

⟨(dexp_{p})_{v}(v), (dexp_{p})_{v}(w)⟩ = ⟨v, w⟩.  (4.4)

Proof. Let w = w_{T} + w_{N} where w_{T} is parallel to v and w_{N} is normal to v. Since dexp_{p} is linear and, by the fact that ||v||^{2} = ||γ′_{v}(t)||^{2} is invariant along γ_{v},

⟨(dexp_{p})_{v}(v), (dexp_{p})_{v}(w_{T})⟩ = ⟨v, w_{T} ⟩,

it is sufficient to prove (4.4) for w = w_{N} ≠ 0.

Since exp_{p}v is defined, there exists ε > 0 such that exp_{p}u is defined for

u = tv(s),  0 ≤ t ≤ 1,  −ε < s < ε,

where v(s) is a curve in T M_{p} with v(0) = v, v′(0) = w_{N} and ||v(s)|| =const. We can, therefore, consider the parameterized surface F : [0, 1] × (−ε, ε) → M given by

(t, s) ↦ F (t, s) = exp_{p}tv(s).

Note that the curves t ↦ F (t, s_{0}) are geodesics on M. Put

F′_{t} = dF (∂/∂t), F′_{s} = dF (∂/∂s).

Clearly,

⟨F′_{t}, F′_{s}⟩(1, 0) = ⟨(dexp_{p})_{v}(v), (dexp_{p})_{v}(w_{N})⟩.

It suffices to prove that

⟨F′_{t}, F′_{s}⟩(1, 0) = 0.

Since t ↦ F (t, ·) are geodesics, ∇_{F′_{t}}F′_{t} = 0. Since ||F′_{t}|| = ||v(s)|| =const. and

[F′_{t}, F′_{s}] = dF [\frac{∂}{∂t}, \frac{∂}{∂s}] = 0,

we have

F′_{t}⟨F′_{t}, F′_{s}⟩ = ⟨F′_{t}, ∇_{F′_{t}}F′_{s}⟩ = ⟨F′_{t}, ∇_{F′_{s}}F′_{t}⟩ = 0,

i.e., ⟨F′_{t}, F′_{s}⟩ is independent of t. Since

lim_{t→0} F′_{s}(t, 0) = lim_{t→0}(dexp_{p})_{tv}tw_{N} = 0,

we conclude that ⟨F′_{t}, F′_{s}⟩(1, 0) = 0. □

Geometrically, Lemma 4.1 is equivalent to the following statement:

Corollary 4.1. Let U_{p} ⊂ M be a neighborhood of p such that exp_{p} is a local diffeomorphism. Then, in U_{p}, the geodesics through p are the orthogonal trajectories of the geodesic hypersurfaces

{exp_{p}(v)|v ∈ T M_{p}, ||v|| = const. }.

2. The normal coordinates. Consider a point p ∈ M and an orhtonormal basis {e_{i}} of T M_{p}. Observing the exponential map exp_{p} is a local diffeomorphism, we take (y^{1}, ⋯, y^{n}) as coordinates of the point q = exp_{p}(y^{i}e_{i}) in a neighborhood U_{p} of p. Such coordinates are called the normal coordinates at p. In the normal coordinate system at p, Γ_{jk}^{i}(p) = 0, because the geodesics that pass through p are given by linear equations. Hence, if X = X^{i} \frac{∂}{∂y^{i}}, then divX(p) = ∑_{i} \frac{∂X^{i}}{∂y^{i}} (p).

Let r^{2} = ∑_{i}(y^{i})^{2}. Taking the polar coordinates (r, θ^{1}, ⋯, θ^{n−1}) in T M_{p}, we obtain the geodesic polar coordinates at p. By the Gauss lemma, ⟨\frac{∂}{∂r} , \frac{∂}{∂θ^{i}} ⟩ = 0, 1 ≤ i ≤ n − 1. For any X = a\frac{∂}{∂r} + ∑ b^{i} \frac{∂}{∂θ^{i}}, we have

⟨\frac{∂}{∂r} , X⟩ = a = X(r),

which implies that the radial function r(q) satisfies

grad r = \frac{∂}{∂r} ,  ⟨\frac{∂}{∂r} , \frac{∂}{∂r} ⟩ = 1.  (4.5)

In the geodesic polar coordinates, the Riemannian metric can be expressed as g = (dr)^{2} + r^{2}h_{ij}(r, θ)dθ^{i}dθ^{j}, 1 ≤ i, j ≤ n − 1.

Lemma 4.2. In U_{p} described as in Corollary 4.1, the shortest path joining two geodesic hypersurfaces around p is a radial geodesic.

Proof. Consider any piecewise smooth curve σ : [a.b] → U_{p} \ {p}. Each point σ(t) can be expressed uniquely in the form exp_{p}(r(t)v(t)) with 0 < r(t) < ε and ||v(t)|| = 1, v ∈ T M_{p}. It suffices that the length ∫_{a}^{b} ||σ′(t)||dt is not less than |r(b) − r(a)|, where equality holds only if the radial function r(t) is monotone and v(t) is constant.

Let F (r, t) = exp_{p}(rv(t)), so that σ(t) = F (r(t), t). Then

σ′(t) = \frac{∂F}{∂r} r′(t) + \frac{∂F}{∂t} .

Since, by the Gauss lemma, the two vector on the right side are mutually orthogonal, and since ||\frac{∂F}{∂r} || = 1, this gives

||σ′(t)||^{2} = |r′(t)|^{2} + ||\frac{∂F}{∂t} ||^{2} ≥ |r′(t)|^{2},

where the equality holds only if ∂F/∂t = 0. Thus,

∫_{a}^{b} ||σ′(t)||dt ≥ ∫_{a}^{b} |r′(t)|dt ≥ |r(b) − r(a)|,

where the equality holds only if r(t) is monotone and v(t) is constant. □

We now show that geodesics locally minimize the arc length.

Proposition 4.1. For any point p ∈ M, there exist a neighborhood W of p and a number δ > 0, such that the length L_{γ}(< δ) of a geodesic segment γ : [0, l] → W is not larger than the length L_{σ} of any piecewise smooth curve σ : [0, l} → W joining γ(0) to γ(l), i.e.,

L_{γ} ≤ L_{σ} with σ(0) = γ(0), σ(l) = γ(l),

where the equality holds if and only if σ = γ.

Proof. Since exp_{p} is a local diffeomorphism, there exist a neighborhood W of p and a constant δ > 0, such that any two points q_{1}, q_{2} ∈ W can be joined by a unique geodesic lying in W whose length is not larger than δ.

Write γ(0) = q_{1} and γ(l) = q_{2}. If σ lies in U_{q_{1}} of q_{1} described as in Lemma 4.2, then we have nothing to do. In the following, we consider the case that does not lie globally in U_{q_{1}}. Set \overline{U}_{q_{1}} = U_{q_{1}} ∪ ∂U_{q_{1}} where ∂U_{q_{1}} is the geodesic hypersurface with radius δ around q_{1}. Let t_{0} ∈ [0, l] such that σ(t_{0}) = q ∈ ∂U_{q_{1}} is the first point, and γ̃ be the radial geodesic from q_{1} to q in \overline{U}_{q_{1}}. Let σ̃ = σ|_{[0,t_{0}]}, so that Lσ̃ ≤ L_{σ}.

By Lemma 4.2, it is clear that Lγ̃ ≤ Lσ̃. On the other hand, since q_{2} ∉ ∂U_{q_{1}}, it is clear that L_{γ} ≤ δ ≤ Lγ̃. Therefor, L_{γ} ≤ L_{σ}. Moreover, the equality holds if and only if γ = σ. □

3. Geodesic completeness. Let (M, g) be a Riemannian manifold, σ : [a, b] → M be a piecewise smooth curve. Then the length of σ is L_{σ} = ∫_{a}^{b} ||σ′(t)||dt. For any two point p, q ∈ M, the distance between p and q is defined by

dist(p, q) = inf{L_{σ}|σ : [a, b] → M, σ(a) = p, σ(b) = q}.

It may be proved that a connected Riemannian manifold is a metric space with the metric dist(,), whose metric topology agrees with the manifold topology. By Proposition 4.1, if a piecewise curve σ from p to q on M has length equal to dist(p, q), then σ is a geodesic.

Definition 4.1. A geodesic segment whose length is the distance between its endpoints is called a minimal geodesic.

Definition 4.2. A Riemannian manifold (M, g) is said to be geodesically complete if the exponential map exp_{p} for a point p ∈ M is a diffeomorphism between T M_{p} and M.

This is clearly equivalent to the following requirement: For each geodesic segment γ_{0} : [a, b] → M it should be possible to extend γ_{0} to an infinite geodesic γ : R → M.

Theorem 4.1 (Hopf and Rinow). Let (M, g) be a connected Riemannian manifold. The following assertions are equivalent:

(i) M is geodesically complete; (ii) M is a complete metric space with metric dist(,); (iii) exp_{p} for any point p ∈ M is defined on all T M_{p}.

One of the above conditions implies that any two points on (M, g) can be joined by a minimal geodesic

For the proof of the theorem, one may be referred to [BS]. Henceforth, we will not distinguish between geodesic completeness and metric completeness, but will refer simply to a complete manifold. Clearly, a compact Riemannian manifold is complete. A closed submanifold of a complete Riemannian is complete in the induced metric.

Note that it is possible that on a complete manifold the minimal geodesic joining two points is not unique. For example, on S^{n} there are infinite minimal geodesics joining two antipodal points.

4. Symmetric spaces. A symmetric space is a connected Riemannian manifold (M, g), such that for each p ∈ M there is an isometry s_{p} : M → M which leaves p fixed and reverses geodesics through p, i.e., if γ is a geodesic and γ(0) = p then s_{p}(γ(t)) = γ(−t). It follows that s_{p} is involutive, i.e., (s_{p})^{2} = Id.

As examples we cite (i) the Euclidean n-space R^{n}, in which case s_{p} is the reflection in p; and (ii) the unit n-sphere S^{n} ⊂ R^{n+1} with the metric induced by R^{n+1}, in which case s_{p} is again reflection in p along a great circle through p.

Proposition 4.2. Let γ be a geodesic in a symmetric space M, and let γ(0) = p and γ(l) = q. Then s_{q}s_{p}(γ(t)) = γ(t + 2l) provided that γ(t) and γ(t + 2l) are defined. Moreover, s_{q}s_{p} preserves parallel vector fields along γ.

Proof. Let γ̃(t) = γ(t + l), so that γ̃ is a geodesic with γ̃(0) = q. Hence,

s_{q}s_{p}(γ(t)) = s_{q}(γ(−t)) = s_{q}(γ̃(−t − l)) = γ̃(t + l) = γ(t + 2l).

Assume that the vector field X(t) is parallel along γ. Then ds_{p}(X) is parallel since s_{p} is an isometry, and ds_{p}(X(0)) = −X(0). Therefore, ds_{p}(X(t)) = −X(t) and hence ds_{q} ∘ ds_{p}(X(t)) = X(t + 2l). □

This proposition shows that geodesics can be infinitely extended, i.e., symmetric spaces are complete.

Corollary 4.2. Let R denote the curvature tensor of a symmetric space M. If X, Y, Z are parallel fields along a geodesic γ, then R(X, Y)Z is also a parallel field along γ.

Proof. Let W be any parallel field along γ. Given p = γ(0) and q = γ(l), consider the isometry J = s_{γ(l/2)}s_{p} which carries p to q. Thus, by Proposition 4.2, we have

⟨R(X_{q}, Y_{q})Z_{q}, W_{q}⟩ = ⟨R(J_{∗}X_{p}, J_{∗}Y_{p})J_{∗}Z_{p}, J_{∗}W_{p}⟩.

Since J is an isometry, this quantity is equal to ⟨R(X_{p}, Y_{p})Z_{p}, W_{p}⟩. Hence, the quantity ⟨R(X, Y)Z, W ⟩ is constant along γ for any parallel field W. It clearly follows that R(X, Y)Z is parallel. □

A Riemannian manifold with ∇R = 0 is called locally symmetric. Corollary 4.2 implies that symmetric spaces are locally symmetric. A classical theorem due to E.Cartan says that a complete, simply connected, locally symmetric space is actually symmetric.

As an example, every compact connected Lie group with the bi-invariant metric is a symmetric.

5. Jacobi fields. Let (M, g) be a complete Riemannian manifold, and γ : [a, b] → M be a differentiable curve with ||γ′(s)|| = l =const. The length of γ is L(γ) = ∫_{a}^{b} ||γ′(s)||ds = l(b − a). A variation of γ is a differentiable map α(s, t) : [a, b] × (−ε, ε) → M such that α(s, 0) = γ(s), α(a, t) = γ(a) and γ(b, t) = γ(b). Set

T := dα(∂/∂s)|_{t=0},  V := dα(∂/∂t)|_{t=0},  γ_{t}(s) := α(s, t)

with V (a, 0) = V (b, 0) = 0. The length of γ_{t}(s) is L(γ_{t}) = ∫_{a}^{b} ||γ′_{t}(s)||ds. Then the first variation formula of arc length is^{[BS]}

\frac{d}{dt}L(γ_{t})|_{t=0} = −l^{−1} ∫_{a}^{b}⟨V, ∇_{T} T ⟩ds.  (4.6)

It shows that a geodesic is the critical point of the arc length functional. The geodesic is normal (namely, the parameter s is its arc length) if l = 1.

Now assume that γ(s) is a normal geodesic. we form the variation of γ as follows: Taking V (0)⊥T and translating parallelly V along σ, we let α(s, t) := exp_{γ(s)}(tV), so that ∇_{V} V = 0 since α(·, t) are geodesics. Such variation α(s, t) consisted of geodesics is called a geodesic variation. For a geodesic variation α(s, t), one can easily see that the second variation formula for γ is^{[BS]}

\begin{aligned}\frac{d^{2}}{dt^{2}} L(γ_{t})|_{t=0} & = ∫_{a}^{b}\{||∇_{T} V ||^{2} − ⟨R(V, T)T, V ⟩\}ds \\ & = − ∫_{a}^{b}⟨∇_{T} ∇_{T} V − R(T, V)T, V ⟩ds := I(V, V), \end{aligned}  (4.7)

where I(V, W) := − ∫_{a}^{b}⟨∇_{T} ∇_{T} − R(T, V)T, W ⟩ds is called the index form, which is a symmetric bilinear form over R defined on the set

V(γ) := {vector field X along γ|X⊥T, X(γ(a)) = X(γ(b)) = 0}.  (4.8)

A vector field V along a geodesic γ : [0, l] → M is called a Jacobi field if it satisfies the following Jacobi equation:

∇_{T} ∇_{T} V − R(T, V)T = 0,  (4.9)

where T = γ′. The following proposition is obvious.

Proposition 4.3. Jacobi fields J along γ have the following properties:

(i) All of Jacobi fields along γ form a 2n-dimensional linear space; (ii) If ⟨J(0), T (0)⟩ = 0, ⟨∇_{T} J(0), T (0)⟩ = 0, then ⟨J, T ⟩ = 0 along γ; (iii) J^{⊥} := J − ⟨J, T ⟩T is also a Jacobi field along γ.

Every Jacobi field along γ : [0, l] → M can be realized by a geodesic variation. In fact, for p = γ(0) and v = γ′(0) = T, we take w ∈ T M_{p} with w⊥v, and extend w to a vector field W along γ. In T M_{p} consider the family of rays:

ρ_{t}(s) := (v + tw)s with ⟨v, w⟩ = 0.

Then

α(s, t) := exp_{p}(ρ_{t}(s)) = exp_{p}((v + tw)s)

is a geodesic variation of γ with T = dα(∂/∂s) |_{t=0} and V = dα(∂/∂s)|_{t=0}. Since ∇_{T} V − ∇_{V} T = dα([∂/∂s, ∂/∂t]) = 0, then

∇_{T} ∇_{T} V = ∇_{T} ∇_{V} T = ∇_{T} ∇_{V} V − ∇_{V} ∇_{T} T = R(T, V)T,

i.e., V is a Jacobi field along γ.

For p ∈ M, let γ : [0, +∞) be a geodesic ray issuing from p = γ(0), which can be expressed by γ(t) = exp_{p}(tv) with v = γ′(0). The point q = γ(t_{0}) is called to be conjugate to p along γ if d(exp_{p})|_{t=t_{0}} = 0, and q is called the first conjugate point to p if there is no any conjugate point in the restriction γ(t)|_{[0,t_{0})}. It is easy to prove^{[BS]}

Proposition 4.4. q is conjugate to p along γ if and only if there exists a Jacobi field J along γ|_{[0,t_{0}]} such that J(p) = J(q) = 0.

Hence, q is conjugate to p if and only if p is conjugate to q.

6. Curvature and topology. By means of the second variation formula (4.7) of geodesics we can prove the following

Theorem 4.2 (Myers and Bonnet). Let (M, g) be an n-dimensional complete Riemannian manifold. If

(i) (Myers) Ric(M) ≥ (n − 1)k for some constant k > 0, or (ii) (Bonnet) K_{M} ≥ k > 0,

then every geodesic of length ≥ π/\sqrt{k} has conjugate points. Hence, the diameter of M satisfies d(M) ≤ π/\sqrt{k}, so that M is actually compact.

Proof. Fix a normalized minimal geodesic γ : [0, l] → M, and let {E_{i}} be an orthonormal basis of parallel fields along γ such that E_{1} = γ′ = T. Let W_{i} = sin(πs/l)E_{i}(t) be vector fields along γ. Then

\begin{aligned}I(W_{i}, W_{i}) & = − ∫_{0}^{l}⟨∇_{T}^{2} W_{i} − R(T, W_{i})T, W_{i}⟩ds \\ & = ∫_{0}^{l}(sin(πs/l))^{2}\{\frac{π^{2}}{l^{2}} + ⟨R(T, E_{i})T, E_{i}⟩\}ds, \end{aligned}

form which it follows that

∑_{i=2}^{n} I(W_{i}, W_{i}) = ∫_{0}^{l}(sin(πs/l))^{2}\{(n − 1)\frac{π^{2}}{l^{2}} − Ric(T, T)\}ds.

So if Ric(M) ≥ (n − 1)k > 0 and l > π/\sqrt{k}, then the sum and therefore at least one summand must be negative. But, this is impossible since some I(W_{i}, W_{i}) < 0 implies that the second variation of arc length is in some direction negative, so that γ cannot be minimal. □

Lemma 4.3 (Jacobi, C.F.). Let γ : [0, l] → M be a normalized geodesic with a conjugate point γ(s_{0}) to γ(0) for s_{0} ∈ (0, l). Then there exists X ∈ V(γ) defined by (4.8) such that I(X, X) < 0.

Proof. Let J be a Jacobi field along γ|_{[0,s_{0}]} with ⟨J, T ⟩ = 0 and J(0) = J(s_{0}) = 0. Then consider the field V along γ defined by

V (s) = J(s)  ∀s ∈ [0, s_{0}],  V (s) = 0  ∀s ∈ [s_{0}, l].

Clearly, V ∈ V(γ) and I(V, V) = 0 by (5.2). Certainly, ∇_{T} J(s_{0}) ≠ 0. Pick Y (s) to be the parallel field along γ for which Y (s_{0}) = −∇_{T} J(s_{0}); let f : [0, l] → R be differentiable such that f(0) = f(l) = 0 and f(s_{0}) = 1 and let

X_{δ} = V + δfY ∀δ > 0.

Then, by a direct calculation and integration by parts, we have

I(X_{δ}, X_{δ}) = I(V, V) + 2δI(V, fY) + O(δ^{2})
= 2δ ∫_{0}^{s_{0}}⟨∇_{T} J, ∇_{T} (fY)⟩ + ⟨R(T, J)T, fY ⟩ + O(δ^{2})
= 2δ⟨∇_{T} J, fY ⟩|_{0}^{s_{0}} + O(δ^{2})
= −2δ||∇_{T} J(s_{0})||^{2} + O(δ^{2}),

which is less than 0 for small δ enough. □

Theorem 4.3 (Cartan-Hadamard). Let (M, g) be a complete Riemannian manifold with nonpositive sectional curvature. Then M has no conjugate points.

Proof. Let γ : [0, l] → M be a normalized geodesic, and X ∈ V(γ). Then, by (4.7),

I(X, X) = ∫_{0}^{l}{||∇_{T} X||^{2} − ||X||^{2}K_{M}(X ∧ T)}ds ≥ 0.

Thus, the claim follows from Lemma 4.3. □

Furthermore, it may be shown that the universal covering space of (M, g) described as in Theorem 4.3 is diffeomorphic to R^{n}. Hence, the homotopy groups π_{i}(M) = 0 for i ≥ 2.^{[BS]}

Chapter 5. Comparison Theorems and Manifold Topology

1. Hessian of the distance function. We first state the following basic index lemma, whose proof can be found in [CE].

Lemma 5.1 (Basic index lemma). Let γ be a geodesic on (M, g) from p to q such that there are no points conjugate to p along γ. Let W be a piecewise smooth vector field on γ and J the unique Jacobi field along γ with J(p) = W (p) = 0 and J(q) = W (q). Then I(J, J) ≤ I(W, W), where the equality holds only if J = W.

Let f : M → R be a differentiable function on M. The second covariant differential ∇df of f is called the Hessian of f denoted by Hessf. For any p ∈ M and X, Y ∈ T M_{p}, we have

Hessf(X, Y) = ∇df(X̃, Ỹ)(p) = Y (X̃f)(p) − (∇_{Y} X̃)f(p),

where X̃ and Ỹ are local extensions of X and Y, respectively. Clearly, Hessf(X, Y) = Hessf(Y, X) and Hessf(X, Y) is independent of the extensions of X and Y. The Laplacian of f is the trace of Hessf (note that there is a sigm difference from the definition as in Ch. 3, §3, the formula (3.15)), i.e.,

Δf = tr(Hessf) = tr(∇^{2}f).

Let (M, g) be a complete manifold, and fix a point o ∈ M. Consider the distance function ρ : M → R defined by

ρ(p) = dist(o, p) ∀p ∈ M, which is a Lipschitz function. The exponential map exp_{o} : T M_{o} → M is defined on T M_{o} since M is complete. For X ∈ T M_{o}, γ(s) = exp_{o}(sX) is a geodesic ray issued from o. Set

s_{X} := sup{s|γ is a unique minimal geodesic joining o and γ(s)}. If s_{X} < +∞, γ(s_{X}) is called a cut point related to o along γ. All of cut points related o is called the cut locus, denoted by cut(o). Clearly, the n-measure of cut(o) is zero. Write μ_{X} := ρ(γ(s_{X})) and put

E := {sX|0 ≤ s < μ_{X}, X ∈ S^{n−1} ⊂ T M_{o}}.

Then exp_{o} : E → M_{o} := exp_{o}(E) ⊂ M is a diffeomorphism and M = M_{o} ∪ cut(o). Clearly, the distance function ρ is smooth in M_{o}, so that the Hessian of ρ is welldefined.

Let (r, θ^{1}, ⋯, θ^{n−1}) be the geodesic polar coordinates at o in M_{o} (cf. Ch. 4, §2), and p = γ(r) ∈ M_{o} for some unique minimal geodesic γ issuing from o. For any X ∈ T M_{p}, we have at p

Hessρ(X, \frac{∂}{∂r}) = Hessr(X, \frac{∂}{∂r})

= X̃(\frac{∂}{∂r} r) − (∇_{X} \frac{∂}{∂r})r

= −⟨∇_{X} \frac{∂}{∂r} , \frac{∂}{∂r} ⟩ = 0.

(5.1)

So, without loss of generality, taking a unit X ∈ T M_{p} satisfying ⟨X, \frac{∂}{∂r} ⟩(p) = 0, we can make a unique Jacobi field J(s) along γ such that (cf. Ch. 4)

J(o) = 0,  J(p) = X,  [J, \frac{∂}{∂r}](s) = 0 ∀0 ≤ s ≤ r.  (5.2)

Thus, at p we have

Hessρ(X, X) = Hessr(X, X) = JJr − (∇_{J}J)r

= J⟨J, \frac{∂}{∂r} ⟩ − ⟨∇_{J}J, \frac{∂}{∂r} ⟩

= ⟨J, ∇_{J} \frac{∂}{∂r} ⟩

= ⟨J, ∇_{∂/∂r}J⟩.

(5.3)

It follows from (5.3) and (4.7) that

\begin{aligned}Hessρ(X, X) & = ∫_{0}^{r} (\frac{d}{ds}⟨J, ∇_{∂/∂s}J⟩) ds \\ & = ∫_{0}^{r}\{||∇_{∂/∂s}J||^{2} − ⟨R(J, \frac{∂}{∂s})\frac{∂}{∂s}, J⟩\}ds = I(J, J). \end{aligned}  (5.4)

Theorem 5.1 (Hessian comparison theorem). Let M_{1} and M_{2} be two ndimensional complete Riemannian manifolds. Assume that γ_{k} : [0, l] → M_{k}, k = 1, 2, are two geodesics parameterized by arc length, and γ_{k} lies in (M_{k})_{γ_{k}(0)}, i.e., γ_{k} does not intersect the cut locus of γ_{k}(0). Let ρ_{k} be the distance function from γ_{k}(0) on M_{k} and let K_{k} be the sectional curvature of M_{k}. Suppose that

K_{1}(Y_{1} ∧ \frac{∂}{∂r_{1}} ) ≥ K_{2}(Y_{2} ∧ \frac{∂}{∂r_{2}} )

along γ_{1}(s) and γ_{2}(s), 0 ≤ s ≤ l, where \frac{∂}{∂r_{k}} = dγ_{k}(\frac{∂}{∂s}), Y_{k} ∈ (T M_{k})_{γ_{k}(s)} with ||Y_{k}|| = 1 and Y_{k}⊥\frac{∂}{∂r_{k}}. Then

Hessρ_{1}(X_{1}, X_{1}) ≤ Hessρ_{2}(X_{2}, X_{2}),

where X_{k} ∈ (T M_{k})_{γ_{k}(l)} with ⟨X_{k}, \frac{∂}{∂r_{k}} ⟩(γ_{k}(l)) = 0 and ||X_{k}|| = 1.

Proof. Let E_{1}^{k}, ⋯, E_{n}^{k} be a system of orthonormal vector fields parallel along γ_{k} with E_{n}^{k} = \frac{∂}{∂r_{k}}, k = 1, 2. By (5.4)

Hessρ_{k}(X_{k}, X_{k}) = I(J_{k}, J_{k}),

where J_{k} is a Jacobi field along γ_{k} with J_{k}(γ_{k}(0)) = 0 and J_{k}(γ_{k}(l)) = X_{k}. Since ⟨X_{k}, \frac{∂}{∂r_{k}} ⟩(γ_{k}(l)) = 0, J_{k} is perpendicular to E_{n}^{k} at each point of γ_{k}. Set

J_{2} = ∑_{i=1}^{n−1} λ_{i}(s)E_{i}^{2} with λ_{i}(0) = 0.

We may take {E_{i}^{1}} so that

X_{1} = J_{1}(γ_{1}(l)) = ∑_{i=1}^{n−1} λ_{i}(l)E_{i}^{1}(γ_{1}(l)).

Define now a vector field Y along γ_{1} by

Y = ∑_{i=1}^{n−1} λ_{i}(s)E_{i}^{1}.

Then Y assumes the same values as J_{1} at s = 0 and s = l. Moreover, ||Y || = ||J_{2}|| and

||∇_{\frac{∂}{∂r_{2}}} J_{2}|| = || ∑_{i=1}^{n−1} λ′_{i}(s)E_{i}^{2}|| = || ∑_{i=1}^{n−1} λ′_{i}(s)E_{i}^{1}|| = ||∇_{\frac{∂}{∂r_{1}}} Y ||.

By Lemma 5.1, we have

Hessρ_{1}(X_{1}, X_{1}) = I(J_{1}, J_{1}) ≤ I(Y, Y)

= ∫_{0}^{l}\{||∇_{\frac{∂}{∂r_{1}}} Y ||^{2} − K_{1}(Y ∧ \frac{∂}{∂r_{1}} )\}ds

= ∫_{0}^{l}\{||∇_{\frac{∂}{∂r_{2}}} J_{2}||^{2} − K_{1}(Y, \frac{∂}{∂r_{1}} )\}ds

≤ ∫_{0}^{l}\{||_{\frac{∂}{∂r_{2}}} J_{2}||^{2} − K_{2}(J_{2}, \frac{∂}{∂r_{2}} )\}ds

= I(J_{2}, J_{2}) = Hessρ_{2}(X_{2}, X_{2}).

□

Corollary 5.1. Under the same hypothesis as in Theorem 5.1, let φ : [0, +∞) → R be a nondecreasing function. Then

Hessφ(ρ_{1})(X_{1}, X_{1}) ≤ Hessφ(ρ_{2})(X_{2}, X_{2}).

2. Laplacian comparison theorems. For later application, we need to compute the hessian of the distance function for space forms. Let \overline{M}(c) be an ndimensional space form of constant curvature c with metric g. In the geodesic polar coordinates (r, θ^{1}, ⋯, θ^{n−1}) at O ∈ \overline{M}(c), g can be expressed by

g = (dr)^{2} + f^{2}(r)dσ^{2},  (5.5)

where dσ^{2} denotes the metric of the unit Euclidean (n−1)-sphere, f(r) is the radial function satisfying the following equation:

f′′ + cf = 0, f(0) = 0, f′(0) = 1,

so that

f(r) = \{ \begin{matrix}sin(\sqrt{c}r)/\sqrt{c}, & ∀c > 0, r < π/\sqrt{c}; \\ r, & ∀c = 0; \\ sh(\sqrt{−c}r)/\sqrt{−c}, & ∀c < 0. \end{matrix}  (5.6)

Let ρ be the distance function from O ∈ \overline{M}(c), and p ∈ \overline{M}(c) which is not conjugate to O. Write ρ(p) = r and take X ∈ T \overline{M}_{p} with ⟨X, \frac{∂}{∂r} ⟩ = 0. Let γ̄ : [0, r] → \overline{M}(c) be the unique minimal normal geodesic joining O and p. It may be proved that the Jacobi field J along γ̄ satisfying (5.2) is^{[BS]}

J(s) = \frac{f(s)}{f (r)}X̃(s),

where f(s) is defined by (5.6) and the vector field X̃ along γ̄ satisfies

∇_{\frac{∂}{∂r}} X̃ = 0,  ⟨X̃, \frac{∂}{∂r} ⟩ = 0,  X̃(r) = X.

By (5.3), we have

Hessρ(X, X) = ⟨J, ∇_{\frac{∂}{∂r}} J⟩|_{p} = ⟨X̃(r), \frac{f′(r)}{f(r)}X̃(r)⟩ = \frac{f′(r)}{f(r)}g(X, X).

Since g(X, \frac{∂}{∂r}) = 0 and Hessρ(X, \frac{∂}{∂r}) = 0, then, noting that ρ = r, we have

Hessρ = \frac{f′(ρ)}{f(ρ)}(g − dρ^{2}) = f(ρ)f′(ρ)dσ^{2},  (5.7)

where f is defined by (5.6). As a consequence of Theorem 5.1, by means of (5.7), we have immediately

Theorem 5.2 (Laplacian comparison theorem). Let M be an n-dimen-sional complete Riemannian manifold, and γ : [0, r] → M be a normal geodesic such that γ(r) is not conjugate to γ(0). Let ρ be the distance function from γ(0) on M. Assume that the Ricci curvature of M satisfies that Ric(γ′) ≥ (n − 1)c, and r < π/\sqrt{c} when c > 0. Then

Δρ(γ(r)) ≤ (n − 1)\frac{f′(r)}{f(r)},  (5.8)

where f is defined by (5.6)

Corollary 5.2. Let M be an n-dimensional complete Riemannian manifold with Ric_{M} ≥ −(n − 1)k, k ≥ 0, and let ρ be any distance function on M. then at any point where ρ is smooth

Δρ ≤ \frac{n − 1}{ρ}(1 + \sqrt{k}ρ).  (5.9)

Proof. By Theorem 5.2 and (5.6) we have

Δρ ≤ (n − 1)\sqrt{k} \frac{ch\sqrt{k}ρ}{sh\sqrt{k}ρ}.

It is easy to check that the function F (x) := (1 + x)shx − xchx is nonnegative, i.e.,

\frac{chx}{shx} ≤ \frac{1 + x}{x}.

□

In particular, if k = 0, then (5.9) holds on M in the sense of distributions,^{[SY]} i.e.,

∫_{M} ρΔφ ≤ ∫_{M} \frac{n − 1}{ρ}φ,  ∀φ ∈ C_{0}^{∞}(M), φ ≥ 0.  (5.10)

3. Volume comparison theorem. Let p ∈ M and let γ(s) be a normalized geodesic with γ(0) = p and γ′(0) = v ∈ T M_{p}. Assume that J(t) is a Jacobi field along γ such that J(0) = 0 and J′(0) = w ∈ T M_{p}. Then

(dexp_{p})_{tv}(v) = γ′(t), (dexp_{p})_{tv}(w) = t^{−1}J(t),

where we have identified T M_{p} with its tangent spaces. If ⟨w, v⟩ = 0, then by the Gauss lemma (lemma 4.1)

⟨(dexp_{p})_{tv}(w), (dexp_{p})_{tv}(v)⟩ = 0.

Let{v = γ′(0), w_{2}, ⋯, w_{n}} be an orthonormal basis in T M_{p}. If γ(r) is not conjugate to p, then let (dexp_{p})_{rv}(v) = γ′(r), (dexp_{p})_{rv}(w_{i}) = r^{−1}J_{i}(r), 2 ≤ i ≤ n, where J_{i} is the Jacobi field along γ with J_{i}(0) = 0 and J′_{i}(0) = w_{i}. Therefore, the Jacobian A(r, θ) of the exponential map at rv is

A(r, θ) = ||γ′(r) ∧ r^{−1}J_{2} ∧ ⋯ ∧ r^{−1}J_{n}||
= r^{1−n}det(J_{2}(r), ⋯ J_{n}(r)),

where θ = (θ^{1}, ⋯, θ^{n−1}) is the spherical component of v in the geodesic polar coordinates {r, θ} at p. By using the above calculation, the following lemma has been proved in [BC]:

Lemma 5.2 (Bishop (i), Günther (ii)). Let (M, g) be an n-dimensional complete Riemannian manifold. (i) If Ric_{M} ≥ (n − 1)c, then r^{n−1}A(r, θ)(f (r))^{1−n} is non-increasing in r, where f(r) is defined by (5.6); (ii) If K_{M} ≤ c, then r^{n−1}A(r, θ)(f(r))^{1−n} is non-decreasing in r.

If we denote the geodesic ball of radius r centered at p by B_{p}(r), then in the geodesic polar coordinates {r, θ}, the volume element of ∂B_{p}(r) is given by r^{n−1}A(r, θ)dθ. On the other hand, (f(r))^{n−1} as in Lemma 5.2 is actually the volume density of the geodesic ball B_{c}(r) of radius r in the space form \overline{M}(c). We will write A_{c}(r) := (f(r))^{n−1} where f is defined by (5.6). Lemma 5.2 then can be restated as

\frac{r^{n−1}A(r, θ)}{A_{c}(r)} is non-increasing in r for Ric_{M} ≥ (n − 1)c;
\frac{r^{n−1}A(r, θ)}{A_{c}(r)} is non-decreasing in r for K_{M} ≤ c.

(5.11)

For c > 0, of course, (5.11) holds only for r ∈ (0, π/\sqrt{c}).

Let S_{r} denote the sphere of radius r centered at the origin in T M_{p}. Then (5.11) implies that

\frac{c^{n−1} ∫_{S_{r}∩E_{p}} A(r, θ)}{∫_{S_{r}} A_{c}(r)} is non-increasing (resp. non-decreasing) in r.  (5.12)

The possible existence of cut locus of p does not present any difficulty here. Set

φ(t) = ∫_{S_{t}∩E_{p}} t^{n−1}A(t, θ), ψ(t) = ∫_{S_{t}} A_{c}(t).

Then

Vol(B_{p}(r)) = ∫_{0}^{r} φ(t)dt, V (c, r) = ∫_{0}^{r} ψ(t)dt,

where V (c, r) denotes the volume of the geodesic ball B_{c}(r) in the space form \overline{M}(c). By Lemma 5.2 and (5.12), it is not hard to prove the following^{[BS]}

Theorem 5.3 (Volume comparison theorem). Let (M, g) be an n-dimen-sional complete Riemannian manifold. For any p ∈ M and r > 0 (r < π/\sqrt{c} for c > 0), (i)(Bishop) if Ric_{M} ≥ (n − 1)c, then Vol(B_{p}(r))/V (c, r) is non-increasing in r. Hence,

Vol(B_{p}(r)) ≤ V (c, r);  (5.13)

(ii)(Günther) if K_{M} ≤ c, then Vol(B_{p}(r))/V (c, r) is non-decreasing in r. Hence,

Vol(B_{p}(r)) ≥ V (c, r).  (5.14)

It should be remarked that the equality of (5.13) hold if and only if B_{p}(r) is isometric to the geodesic ball B_{c}(r) of radius r in \overline{M}(c), which was shown in [BC]. So, combining with the Bonnet-Myers theorem (cf. Ch. 4, §6), we have

Theorem 5.4 (Toponogov and S.Y.Cheng). Let (M, g) be an n-dimensional complete Riemannian manifold with Ric_{M} ≥ (n − 1)c > 0. If the diameter of M is equal to π/\sqrt{c}, then (M, g) is isometric to the standard n-sphere of constant sectional curvature c.

Proof (Shiohama, 1983). By the Bonnet-Myers theorem, (M, g) is actually compact. Pick points p, q ∈ M so that dist(p, q) = π/\sqrt{c}. Then Theorem 5.3 (i) implies that

\frac{Vol(B_{p}(\frac{π}{2\sqrt{c}}))}{V (c, \frac{π}{2\sqrt{c}})} ≥ \frac{Vol(B_{p}(\frac{π}{\sqrt{c}}))}{V (\overline{M}(c))} = \frac{Vol(M)}{V (\overline{M}(c))}.

Therefore,

Vol(B_{p}(\frac{π}{2\sqrt{c}})) ≥ \frac{Vol(M)}{2}.

Similarly, we have also

Vol(B_{q}(\frac{π}{2\sqrt{c}})) ≤ \frac{Vol(M)}{2}.

But

Vol(B_{p}(\frac{π}{2\sqrt{c}}) ∩ B_{q}(\frac{π}{2\sqrt{c}})) = 0.

Therefore,

Vol(B_{p}(\frac{π}{2\sqrt{c}})) = Vol(B_{q}(\frac{π}{2\sqrt{c}})) = \frac{Vol(M)}{2}.

Hence, Vol(M) = V (\overline{M}(c)), and M is isometric to \overline{M}(c). □

4. Splitting theorem. A geodesic line (resp. geodesic ray) γ on a complete Riemannian manifold (M, g), is a minimal normalized geodesic γ : (−∞, +∞) → M (resp. [0, +∞) → M), i.e.,

d(γ(t_{1}), γ(t_{2})) = dist(γ(t_{1}), γ(t_{2})) = |t_{1} − t_{2}|,
∀t_{1}, t_{2} ∈ (−∞, +∞) (resp. [0, +∞)).

If M is complete and non-compact, then for any o ∈ M there is a geodesic ray starting from o.

Suppose that γ is a geodesic ray on M; then the Busemann function associated with γ is defined by

B^{+}(p) = lim_{t→∞} B_{t}^{+} := lim_{t→∞}(d(p, γ(t)) − t) ∀p ∈ M.

Then we have by the triangle inequality

|B_{t}^{+}(p)| ≤ d(γ(0), p).

Hence, the family {B_{t}^{+}(p)}_{t≥0} is uniformly bounded on any compact subset of M. Moreover, if s < t then

B_{t}^{+}(p) − B_{s}^{+}(p) = d(p, γ(t)) − d(p, γ(s)) − d(γ(s), γ(t)) ≤ 0.

Thus, B_{t}^{+}(p) is monotone decreasing. It follows that B_{t}^{+}(p) → B^{+}(p) as t → ∞, uniformly on any compact subset.

On the other hand, since |B_{t}^{+}(p) − B_{t}^{+}(q)| ≤ d(p, q) for any t ≥ 0, we have (B^{+}(p) − B^{+}(q)| ≤ d(p, q). This means that B^{+}(p) is Lipschitz continuous, so it is differentiable almost everywhere.

By definition, the set {p ∈ M|B_{t}^{+}(p) < C} is just a geodesic ball centered at γ(t). Therefore, we may consider intuitively the set {p ∈ M|B^{+}(p) < C} as a geodesic ball centered at γ(∞).

If γ is a geodesic line, we may define similarly another Busemann function B^{−}(p) by

B^{−}(p) = lim_{t→−∞}(d(p, γ(t)) + t).

It is easy to check that

B^{+}(p) + B^{−}(p) ≥ 0  ∀p ∈ M,  (5.15)

where the equality holds if p ∈ γ. Now we can prove the following

Theorem 5.5 (Cheeger-Gromoll). Let M be an n-dimensional complete Riemannian manifold with Ric_{M} ≥ 0. If M has a geodesic line, then M is isometric to R × N, where N is an (n − 1)-dimensional Riemannian manifold and R × N has product metric.

Proof. Let γ be the geodesic line in M and B^{+}, B^{−} be the Busemann functions associated with γ. By (5.10), for any φ ∈ C_{0}^{∞}(M), φ ≥ 0,

∫_{M} B_{t}^{+}Δφ = ∫_{M} φΔB_{t}^{+} = ∫_{M} φΔd(p, γ(t)) ≤ ∫ \frac{n − 1}{d(p, γ(t))}φ.

Letting t → ∞ we get

∫_{M} B^{+}Δφ ≤ 0,

which means that ΔB^{+} ≤ 0 in the sense of distributions. Similarly, we have ΔB^{−} ≤ 0. Hence,

Δ(B^{+} + B^{−}) ≤ 0.

It follows from the mean value inequality that^{[SY]}

B^{+}(p) + B^{−}(p) ≥ \frac{1}{ω_{n−1}r^{n}} ∫_{B_{p}(r)}(B^{+} + B^{−}),

where B_{p}(r) is the geodesic ball in M of radius r centered at p, and ω_{n−1} stands for the volume of the standard unit (n−1)-sphere. Now if p ∈ γ, we see from (5.15) that B^{+} + B^{−} ≡ 0 on B_{p}(r). This shows that B^{+} + B^{−} is identically zero in a neighborhood of γ. Then, applying the mean value inequality repeatedly, one can easily see

B^{+} + B^{−} ≡ 0 on M.

It follows that ΔB^{+} + ΔB^{−} = 0. Since ΔB^{+} ≤ 0 and ΔB^{−} ≤ 0, then

ΔB^{+} = 0, ΔB^{−} = 0

in the sense of distributions. By Weyl’s lemma in the theory of linear elliptic PDE, B^{+} and B^{−} are actually smooth. On the other hand, for the distance function ρ, ||∇ρ|| = 1 almost everywhere implies that ||∇B^{+}|| = 1. Since Ric_{M} ≥ 0, a simple calculation shows that ∇(∇B^{+}) = 0, i.e., ∇B^{+} is a parallel field on M. By the de Rham decomposition theorem, the flow generated by ∇B^{+} (i.e., the gradient flow of B^{+}) can be used to construct an isometric diffeomorphism from R × N onto M, where N = {p ∈ M|B^{+}(p) = 0}. This completes the proof of the theorem. □

By induction, it is not hard to see the following

Corollary 5.3. Let M be an n-dimensional complete Riemannian manifold with Ric_{M} ≥ 0. Then M is isometric to R^{k} ×N, where N does not contain any geodesic lines.

Using this result we can obtain a structure theorem for the fundamental group of compact manifolds (cf. [SY]).

Corollary 5.4. Let M be a compact Riemannian manifold with non-negative Ricci curvature. Then π_{1}(M) contain a normal subgroup H such that π_{1}(M)/H is finite. Moreover, the universal covering M̃ of M is isometric to R^{k} × M̄ where M̄ is compact.

Chapter 6. Complex Manifolds

1. Complex manifolds. A complex manifold of complex dimension n is a Hausdroff topological space M such that there is given a family of {(U_{α}, ϕ_{α})} with ⋃_{α} U_{α} = M, where ϕ_{α} is a homeomorphism of U_{α} onto an open set of ℂ^{n} and ϕ_{β} ∘ ϕ_{α}^{−1} is holomorphic on ϕ_{α}(U_{α} ∩ U_{β}). We can attach the complex coordinate of the point ϕ_{α}(q) ∈ ϕ_{α}(U_{α}) in ℂ^{n} to the point q ∈ U_{α} ⊂ M, often written z = (z_{1}, ⋯, z_{n}).

Example 1. A 1-dimensional complex manifold is called a Riemannian surface (cf. Ch. 1, §1).

Example 2. Let ℂP^{n} denote the set of (complex) lines through the origin of ℂ^{n+1}. A line l in ℂ^{n+1} is determined by any z = (z_{0}, ⋯, z_{n}) ≠ 0 ∈ l, so we can write

ℂP^{n} = {[z] ≠ 0 ∈ ℂ^{n+1}}/{[z] ∼ [λz]}.

On the subset U_{i} = {[z]|z_{i} ≠ 0} ⊂ ℂP^{n} of lines not contained in the hyperplane (z_{i} = 0), there is a bijective map ϕ_{i} to ℂ^{n} given by

ϕ_{i}([z_{0}, ⋯, z_{n}]) = (\frac{z_{0}}{z_{i}} , ⋯, \frac{z_{i−1}}{z_{i}} , \frac{ẑ_{i}}{z_{i}} , ⋯, \frac{z_{n}}{z_{i}} ) = (_{i}ζ^{k})_{k≠i},

where we have set _{i}ζ^{k} := \frac{z_{k}}{z_{i}}. On (z_{i} ≠ 0) = ϕ_{i}(U_{j} ∩ U_{i}) ⊂ ℂ^{n},

ϕ_{j} ∘ ϕ_{i}^{−1}(_{i}ζ^{k}) = (_{i}ζ^{k}/_{i}ζ^{j})_{k≠j} = (_{j}ζ^{k})_{k≠j}

is clear holomorphic; thus ℂP^{n} is a complex manifold, called complex projective space. (z_{0}, ⋯, z_{n}) is called the homogeneous coordinates of ℂP^{n} and (_{i}ζ^{k})_{k≠i} is called the inhomogeneous coordinates. Note that ℂP^{1} is just the Riemann sphere ℂ ∪ {∞}.

By assigning to a point (z_{0}, ⋯, z_{n}) of ℂ^{n+1} \ {0} the point [z_{0}. · · · , z_{n}] ∈ ℂP^{n}, we obtain a natural projection π : ℂ^{n+1} \ {0} → ℂP^{n}, for which the inverse image of each point is ℂ^{∗} = ℂ^{1} \ {0} called the multiplicative group of nonzero complex numbers. Then, ℂ^{n+1} \ {0} is a principle fibre bundle over ℂP^{n} with group ℂ^{∗}.

The homogeneous coordinates of ℂP^{n} can be normalized so that

∑_{k=0}^{n} z_{k}z̄_{k} = 1,

which defines a unit sphere S^{2n+1} in ℂ^{n+1} ≡ R^{2n+2}. The restriction of πto S^{n+1} gives the projection π : S^{n+1} → ℂP^{n}, under which the inverse image of each point is a circle S^{1} called the multiplicative group of complex numbers of absolute value 1. Thus, S^{2n+1} → ℂP^{n} a principle fibre bundle with group S^{1}. This is called the Hopf fibering of S^{2n+1}. Note that the group S^{1} may be identified with the additive group R/ℤ (the real numbers modulo 1); the isomorphism is given by λ = e^{2πiθ} → θ = lnλ/2πi ∈ R/ℤ. Thus, S^{2n+1} → ℂP^{n} can be viewed as the principle fibre bundle with group R/ℤ.

More general, as in the real case (cf. Ch. 1, §1), we can consider the complex Grassmannian G_{k,n}(ℂ) := U(n)/U(k) × U(n − k) where U(n) denotes the unitary group of order n. Thus, ℂP^{n} = G_{1,n+1}(ℂ).

Example 3. Let Λ = ℤ^{k} ⊂ ℂ^{n} be a discrete lattice. Then the quotient group ℂ^{n}/Λ has the structure of a complex manifold induced by the projection map π : ℂ^{n} → ℂ^{n}/Λ. It is compact if and only if k = 2n; in this case ℂ^{n}/Λ is called a complex torus.

Example 4. A Hopf surface is defined to be the quotient of ℂ^{2} \ {0} by the group of automorphisms generated by z ↦ 2z. This is the simplest example of a compact complex manifold which cannot be imbedded in any ℂP^{n}.

Let M be a complex manifold of complex dimension n, and (U, ϕ) be a neighborhood with complex coordinates \{z_{j}\}. We write z_{j} = x_{j} + iy_{j} with i = \sqrt{−1} so that M can be considered as a smooth real manifold of dimension 2n. For any point p ∈ U ⊂ M, the tangent space T M_{p} has a natural structure of a complex vector space as follows: Consider ℂ^{n} ≃ R^{2n} by the isomorphism

(z_{1}, ⋯, z_{n}) ↦ (x_{1}, y_{1}, ⋯, x_{n}, y_{n}).

Since dz_{j} = dx_{j} + idy_{j}, then its dual complex vector is naturally

\frac{∂}{∂z_{j}} = \frac{1}{2}(\frac{∂}{∂x_{j}} − i\frac{∂}{∂y_{j}}).

Define the automorphism J on T M_{p} by

J(\frac{∂}{∂x_{j}}) = \frac{∂}{∂y_{j}} ,  J(\frac{∂}{∂y_{j}}) = −\frac{∂}{∂x_{j}} .  (6.1)

It turns out that dz_{j}(\frac{∂}{∂x_{k}}) = δ_{k}^{j} and dz_{j}(J \frac{∂}{∂x_{k}}) = iδ_{k}^{j}, so that \{\frac{∂}{∂x_{j}} \}_{p} as well as \{\frac{∂}{∂y_{j}} \}_{p} form a ℂ-basis of T M_{p}. Therefore, (6.1) determines an automorphism on the tangent bundle T M satisfying J^{2} = −Id. It is called the complex structure on T M.

2. Almost complex structures. An almost complex structure on a (real) manifold M is a differentiable (1, 1)-tensor field J of automorphisms of the tangent bundle T M satisfying

J_{p}^{2} = −Id_{p}, p ∈ M

where Id denotes the identity. Equivalently, an almost complex structure on M is an action of the field ℂ of complex numbers on T M giving to it a structure of complex vector bundle over M. In such a case M is called a almost complex manifold.

Clearly, the dimension of a almost complex manifold M is even, 2n, where n is the complex dimension of M. Moreover, the almost complex structure J induces a preferred orientation on M, for which adapted orthonormal frames {e_{1}, Je_{1}, ⋯, e_{n}, Je_{n}} are positively oriented.

The almost complex structure J induces a splitting of the complexified tangent bundle T^{ℂ}M = T M ⊗_{R} ℂ into two complementary complex subbundles, conjugate to each other:

T^{ℂ}M = T′M ⊕ T′′M,  (6.2)

where at each point p ∈ M, the fiber T′M_{p} (resp. T′′M_{p}) is the eigenspace of J_{p} relative to the eigenvalue +i (resp. −i). Conversely, each complex subbundle T′M of T^{ℂ}M with complementary conjugate determines an almost structure J on M.

Clearly, the complex structure (6.1) on a complex manifold (veiwed as a real manifold) is an almost complex structure. Conversely, an almost complex structure J on a manifold M is called a complex structure if M is an underlying manifold of a complex manifold which induces J in the way described as in (6.1). The question arises whether every almost complex structure is complex. This is the case for n = 2, but not in general.

Let J be an almost complex structure on a manifold M of real dimension 2n. We define the torsion of J to be the (1, 2)-tensor field N given by

N(X, Y) = 2{[JX, JY] − [X, Y] − J[X, JY] − J[JX, Y]}  (6.3)

for X, Y ∈ C(T M). An almost complex structure is said to be integrable if it has no torsion.

Theorem 6.1 (Newlander-Nirenberg). An almost complex structure is complex if and only if it is integrable, i.e., it has no torsion.

Proof. We shall only prove here that a complex structure has no torsion. The converse may be found in the Newlander-Nirenberg paper (Ann. of Math., 65(1957), 391-404).

Let {z_{k} = x_{k} + iy_{k}} be the local complex coordinates on a complex manifold M. Writing y_{k} = x_{n+k}, we have the local real coordinates {x_{α}} on M, 1 ≤ α, β, ⋯ ≤ 2n. On putting

J(∂_{α}) = J_{α}^{β}∂_{β},  N(∂_{α}, ∂_{β}) = N_{αβ}^{γ}∂_{γ},  ∂_{α} = \frac{∂}{∂x_{α}} ,

we have by (6.3)

N_{αβ}^{γ} = 2(J_{α}^{σ}∂_{σ}J_{β}^{γ} − J_{β}^{σ}∂_{σ}J_{α}^{γ} − J_{σ}^{γ}∂_{α}J_{β}^{σ} + J_{σ}^{γ}∂_{β}J_{α}^{σ}).

From (6.1) i follows that the components J_{α}^{β} of the complex structure J are constant. Hence, N_{αβ}^{γ} = 0 so that N = 0. □

We now consider the splitting (6.2). The elements of T′M (resp. T′′M) are called the (complex) vector fields of type (1, 0) (resp. of type (0, 1)). Each real tangent vector X on M can be expressed uniquely as

X = U + Ū,

where U (resp. Ū) is its part of type (1, 0) (resp. of type (0, 1)). The complex components U and Ū are, in turn, determined by

U = \frac{1}{2}(X − iJX),  Ū = \frac{1}{2}(X + iJX).

We have, in this way, a ℂ-linear isomorphism of T M onto T′M and a skew ℂ-linear isomorphism of T M onto T′′M.

The above splitting (6.2) of T^{ℂ}M induces a splitting into types of whole complex tensor bundle. In particular, we have

Λ_{ℂ}^{r}(M) = ∑_{p+q=r} Λ^{p}(T′M)^{∗} ⊗ Λ^{q}(T′′M)^{∗} := Λ_{ℂ}^{p,q}(M),  (6.3)

where Λ_{ℂ}^{r}(M) = Λ^{r}(M) ⊗ ℂ is the bundle of the ℂ-valued r-forms, and Λ^{p}(T′M)^{∗} (resp. Λ^{q}(T′′M)^{∗}) denotes the bundle of the ℂ-linear p-forms (resp. q-forms) on T′M (resp. T′′M). The elements of Λ_{ℂ}^{p,q}(M) are called (complex) forms of type (p, q). The following properties are easily verified:

(1) If θ ∈ Λ_{ℂ}^{(p,q)}(M), then θ̄ ∈ Λ_{ℂ}^{(q,p)}(M), so that forms of type (p, p) are real;

(2) If α ∈ Λ_{ℂ}^{(p,q)}(M) and β ∈ Λ_{ℂ}^{(r,s)}(M), then α ∧ β ∈ Λ_{ℂ}^{(p+r,q+s)}(M);

(3) dΛ_{ℂ}^{(p,q)}(M) ⊂ Λ_{ℂ}^{(p+2,q−1)}(M)⊕Λ_{ℂ}^{(p+1,q)}(M)⊕Λ_{ℂ}^{(p,q+1)}(M)⊕Λ_{ℂ}^{(p−1,q+2)}(M);

(4) Λ_{ℂ}^{(p,q)} = 0 if p > n or q > n. In fact, the complex operator J induces a complex operator, also denoted by J, on the cotangent bundle T M^{∗} = Λ^{1}(M), defined by

Jθ(X) := −θ(JX)  for θ ∈ T M_{p}^{∗}, X ∈ T M_{p}, p ∈ M.  (6.4)

This operator J extends to the whole Λ_{ℂ}^{∗}(M); we have

Jφ = i^{(q−p)}φ  ∀φ ∈ Λ_{ℂ}^{p,q}(M).  (6.5)

Proposition 6.1. For an almost complex manifold M, the following conditions are equivalent:

(1) The almost complex structure is integrable; (2) The Lie bracket of any two vector fields of type (1, 0) (resp. (0, 1)) is

likewise a vector field of type (1, 0) (resp. (0, 1));

(3) For any θ ∈ Λ^{(p,q)}(M), dθ ∈ Λ^{(p+1,q)}(M) ⊕ Λ^{(p,q+1)}(M).

Proof. To prove (2) ⇔ (3), let θ ∈ Λ_{ℂ}^{(1,0)}(M). If Z, W ∈ C(T′′M), then (2) implies that

2dθ(U, W) = U(θ(W)) − W (θ(U)) − θ([U, W]) = 0,  (6.6)

which shows that dθ does not contain the component of type (0, 2). Similarly, if θ ∈ Λ_{ℂ}^{(0,1)}(M), then (2) implies that dθ does not contain thr component of type (2, 0). Since Λ_{ℂ}^{∗}(M) is locally generated by Λ_{ℂ}^{(0,0)}(M), Λ_{ℂ}^{(1,0)}(M) and Λ_{ℂ}^{(0,1)}(M), this shows (3). Conversely, (3) and (6.6) shows that θ([U, W]) = 0 for all θ ∈ Λ_{ℂ}^{(1,0)}(M). Hence, [U, W] ∈ C(T′′M). This shows (2).

To prove (1) ⇔ (2), let X, Y be (real) vector fields and let Z = [X−iJX, Y −iJY]. Then (2) holds if and only if Z ∈ C(T′M) for all X, Y. On the other hand, a simple calculation shows that

2(Z + iJZ) = −N(X, Y) − iJ(N(X, Y)).

It is obvious that Z + iJZ = 0 if and only if Z ∈ C(T′M). Hence (2) holds if and only if N(X, Y) = 0 for all X, Y, i.e., by Theorem 6.1, J is integrable. □

When J is integrable, the exterior differential dθ of any form θ of type (p, q) is the sum of a form of type (p+1, q) and a form of type (p, q+1) denoted respectively by ∂θ and ¯∂θ. Thus, we define two differential operators:

∂ : Λ_{ℂ}^{(p,q)}(M) → Λ_{ℂ}^{(p+1,q)}(M),  ¯∂: Λ_{ℂ}^{(p,q)}(M) → Λ_{ℂ}^{(p,q+1)}(M),

so that

d = ∂ + ¯∂,

which satisfy that

∂^{2} = 0,  ¯∂^{2} = 0,  ∂ ¯∂ + ¯∂∂ = 0.  (6.7)

We can define a new (real) differential operator d^{c}, acting on forms, by

d^{c}φ = −J^{−1}dJφ = (−1)^{r}JdJφ  (6.8)

for any r-form φ. We have

d^{c} = i(¯∂ − ∂),

and equivalently,

∂ = \frac{1}{2}(d + id^{c}),  ¯∂ = \frac{1}{2}(d − id^{c}).  (6.9)

It is easy to see that

(d^{c})^{2} = 0,  dd^{c} + d^{c}d = 0,  dd^{c} = 2i∂ ¯∂.  (6.10)

Remark. In some books, the symbols ∂ and ¯∂ are replaced by d′ and d′′, respectively.

A form θ ∈ Λ_{ℂ}^{(p,q)}(M) is called ¯∂-closed if ¯∂θ = 0. Let Z_{d′′}^{(p,q)}(M) denote the space of all ¯∂-closed forms of type (p, q). The quotient groups

D^{(p,q)}(M) := Z(p,q) ; ¯∂(M)/¯∂Λ_{ℂ}^{(p,q−1)}(M)

are called the Dolbeault cohomology groups of M, which are analogous to the de Rahm cohomology groups of a real manifold (cf. Ch. 3). In ℂ^{n} with the complex coordinates {z_{k}}, let D be a smaller polydisc |z_{k}| < c_{k}. Let α ∈ Λ_{ℂ}^{(p,q)}(D) be ¯∂-closed. Then there exists a form β ∈ Λ_{ℂ}^{(p,q−1)}(D) such that ¯∂β = α in D′ ⊂ D. This is the so-called Dolbeault-Grothendieck lemma.^{[Ch]}

3. Hermitian and Kähler metrics. From now on, let M be a complex manifold or, equivalently, an almost complex manifold with an integrable almost structure J. A Hermitian metric on M is a Riemannian metric g satisfying

g(Jx, JY) = g(X, Y)  ∀X, Y ∈ C(T M).  (6.11)

Equivalently, the metric is J-invariant. A Hermitian metric is Kähler if the complex structure J is parallel with respect to the Levi-Civita connection ∇, i.e.,

∇J = 0 ⇔ ∇_{X}JY = J∇_{X}Y ∀X, Y ∈ C(T M).  (6.12)

In other words, the Hermitian condition means that J_{p} is an isometry of T M_{p} for each p ∈ M. The Kähler condition means that the tensor field J is invariant under parallelism.

We denote by the same symbols the various tools of Riemannian geometry and their ℂ-linear or multilinear extensions to the complexified bundles. Then for the Kähler metric g we have

R(X, Y) = ∇_{X}∇_{Y} − ∇_{Y} ∇_{X} − ∇_{[X,Y]}

for any X, Y ∈ T^{ℂ}M.

Proposition 6.2. Let M be a complex manifold with complex structure J and Riemannian metric g.

(1) If g is Hermitian, then g(X, Y) = 0 for X, Y ∈ C(T′M) (or C(T′′M));

(2) If g is Kähler and R denotes the curvature tensor, then R(X, Y) = 0 for X, Y ∈ C(T′M) (or C(T′′M)).

Proof. If g is Hermitian and X, Y ∈ C(T′M), then

g(X, Y) = g(JX, JY) = g(iX, iY) = −g(X, Y),

which yields that g(X, Y) = 0.

For any complex vector fields Z, W on M, the Kähler condition (6.12) implies that R(Z, W)X is type (1, 0) as well as X. Hence, by above (1),

g(R(Z, W)X, Y) = 0.

By the algebraic properties of R (cf. Ch. 2, §3), the above condition is equivalent to

g(R(X, Y)Z, W) = 0.

Since Z, W are arbitrary and g is nondegenerate for each p ∈ M, the lemma follows. □

The Kähler form ω of a Hermitian metric on a complex manifold M is the 2-form of type (1, 1) defined (up to a constant factor) by

ω(X, Y) := g(X, JY) ∀X, Y ∈ C(T^{ℂ}M). (6.13)

Since g is invariant by J, so is ω, i.e., ω(JX, JY) = ω(X, Y).

Proposition 6.3. A Hermitian metric is Kähler if and only if the Kähler form is closed.

Proof. If g is Kähler, then g and J are parallel so that ω defined by (6.13) is parallel. From the formula (3.4) of Ch. 3, §2. it follows that ω is closed. Conversely, since J has no torsion, we have (cf. [KN], p.148)

4g((∇_{X}J)Y, Z) = 6dω(X, JY, JZ) − 6dω(X, Y, Z).

Hence, if dω = 0, then ∇_{X}J = 0 so that g is Kähler. □

Example 1. The Hermitian metric on ℂ^{n} given by

g = ∑_{j=1}^{n} dz_{j} ⊗ dz̄_{j}  (6.14)

is called the Euclidean or standard metric. The induced Riemannian metric is, of course, the standard metric on ℂ^{n} = R^{2n}. The Kähler form is

ω = −i ∑_{j=1}^{n} dz_{j} ∧ dz̄_{j},

which is closed obviously.

If Λ ⊂ ℂ^{n} is a full lattice, then the metric given on the complex torus ℂ^{n}/Λ by (6.14) is again called the Euclidean metric on ℂ^{n}/Λ.

Example 2. Let (z_{0}, ⋯, z_{n}) be coordinates in ℂ^{n+1} and denote by π : ℂ^{n+1} \ {0} → ℂP^{n} the natural projection (cf. §1, Ex.2). Let U ⊂ ℂP^{n} be an open set and z : U → ℂ^{n+1} \ {0} a lifting of U, i.e., a holomorphic map with π ∘ z = Id; consider the following 2-form

ω = −4i∂ ¯∂log |z|^{2}.

If z′ : U → ℂ^{n+1} \ {0} is another lifting, then z′ = f · z with a nonzero holomorphic function f (i.e., ¯∂f = 0), so that

−4i∂ ¯∂log |z′|^{2} = −4i∂ ¯∂(log |z|^{2} + log f + log f̄)
= ω − 4i(∂ ¯∂log f − ¯∂∂ log f̄) = ω.

Therefore, ω is independent of the lifting chosen; since liftings always exist locally, ω is a globally defined 2-form in ℂP^{n}. Clearly ω is of type (1, 1). To see that ω is positive, first note that the unitary group U(n + 1) acts transitively on ℂP^{n} and leaves the form ω invariant, so that ω is positive everywhere if it is positive at one point. Now let {w_{j} = z_{j}/z_{0}} be homogeneous coordinates on the open set U_{0} (z_{0} ≠ 0) in ℂP^{n} and use the lifting z = (1, w_{1}, ⋯, w_{n}) on U_{0}; we have

\begin{aligned}ω & = −4i∂ ¯∂log(1 + ∑ w_{j}w̄_{j}) = −4i∂ (\frac{∑ w_{j}dw̄_{j}}{1 + ∑ w_{j}w̄_{j}} ) \\ & = −4i (\frac{∑ dw_{j} ∧ dw̄_{j}}{1 + ∑ w_{j}w̄_{j}} − \frac{(∑ w̄_{j}dw_{j}) ∧ (∑ w_{j}dw̄_{j})}{(1 + ∑ w_{j}w̄_{j})^{2}}) . \end{aligned}

At the point (1, 0, ⋯, 0),

ω = −4i ∑ dw_{j} ∧ dw̄_{j} > 0.

By (6.13), ω defines a Hermitian metric on ℂP^{n}

g = 4\frac{(1 + ∑ w_{j}w̄_{j})(∑ dw_{j} ⊗ dw̄_{j}) − (∑ w̄_{j}dw_{j}) ⊗ (∑ w_{j}dw̄_{j})}{(1 + ∑ w_{j}w̄_{j})^{2}},  (6.15)

called the Fubini-Study metric. Clearly, this metric is Kähler.

Example 3. Let Δ be the discontinuous group generated by z_{k} → 2z_{k}, ! ≤ k ≤ n. The quotient manifold (ℂ^{n} \ {0})/Δ is called the Hopf manifold. It is homeomorphic to S^{1} × S^{2n−1}.

Let p, q > 0 be integers. By the Hopf fibering (§1, Ex. 2), we have a fiberatio

π : S^{2p+1} × S^{2q+1} → ℂP^{p} × ℂP^{q}

with fiber a two (real) dimensional torus. We can give the total space S^{2p+1}×S^{2q+1} a complex structure J so that S^{2p+1} × S^{2q+1} becomes a complex manifold, called the Calabi-Eckmann manifold.^{[Ch]}

It has been pointed out in ([Ch], p.59) that The Hopf and Calabi-Eckmann manifolds can not be given a Kähler structure since compact Kähler manifolds have strong topological restrictions.

4. Hermitian symmetric spaces. Let M be a connected complex manifold with a Hermitian structure. The set of holomorphis isometries of M forms a group, denoted by A(M), the group operation being composition of mappings. M is said to be a Hermitian symmetric space if each point p ∈ M is an isolated fixed point of an involutive holomorphic isometry s_{p} of M, i.e., s_{p}^{2} = Id. A Hermitian symmetric space M is of course a Riemannian symmetric space of even dimension (cf. Ch. 4, §4). Hence the isometric group I(M) has a Lie group structure compatible with the compact open topology and is a Lie transformation group of M. The group A(M) is a closed subgroup of I(M) and is therefore also a Lie transformation group of M. It is transitive on M since it contains all the symmetries. The identity component G ≡ A_{0}(M) of A(M) is also transitive on M (cf. [He], Ch. II, Prop. 4.3). Let o ∈ M and let K be the subgroup of G leaving o fixed. With the automorphisim a → s_{o}as_{o} of G for a ∈ G, the pair (G, K) is a Riemannian symmetric pair, i.e., there exists an involutive automorphism σ ∈ G such that (K_{σ})_{0} ⊂ K ⊂ K_{σ} and Ad_{G}(K) is compact, where K_{σ} is the set of fixed points of σ and (K_{σ})_{0} is the identity component of K_{σ}. Thus, M is diffeomorphic to G/K.^{[He]}

Let X ∈ T M_{o} and let γ_{X}(t) denote the geodesic on M with γ′(0) = X. Let s_{t} denote the geodesic symmetry (extended to M) with respect to the point γ_{X}(t). We see that if T_{t} = s_{t/2}s_{o} then t → T_{t} is a one-parameter subgroup of G (cf. [He], Ch. IV, §3), T_{t} ·o = γ_{X}(t) and (dT_{t})_{o} is the parallel translation along γ_{X}. Since the elements of G by definition leave the complex structure of M invariant, the same is true of the parallel translation (dT_{t})_{o}. This proves the following

Proposition 6.4. The Hermitian structure of a Hermitian symmetric space is Kähler.

Recall that a Hermitian symmetric space M is said to be of the compact type or the noncompact type according to the type of the Riemannian symmetric pair (A_{o}(M), K), K being the isotropy subgroup of A_{o}(M) at some point o ∈ M.^{[He]} It is proved in ([He], Ch. VIII, §4) that any Hermitian symmetric space of the compact type or the noncompact type is simply connected.

Chapter 7. Hermitian and Kähler Geometry

1. Ricci form. Let M be a complex manifold with complex dimension n. Let {z_{1}, ⋯, z_{n}} be the local complex coordinates in an open neighborhood U of a point p ∈ M. If we write z_{j} = x_{j} + iy_{j}, 1 ≤ j, k ≤ n, then {x_{1}, y_{1}, ⋯, x_{n}, y_{n}} is a real coordinate system on the underlying analytic manifold U ⊂ M. The vector fields given by

Z_{j} := \frac{∂}{∂z_{j}} = \frac{1}{2}(\frac{∂}{∂x_{j}} − i\frac{∂}{∂y_{j}}), Zj̄ := \frac{∂}{∂z̄_{j}} = \frac{1}{2}(\frac{∂}{∂x_{j}} + i\frac{∂}{∂y_{j}})

are of type (1, 0) and (0, 1), respectively. Their dual 1-forms are respectively

dz_{j} = dx_{j} + idy_{j}, dz̄_{j} = dx_{j} − idy_{j}.

A function f on U is holomorphic if and only if ∂f/∂z̄_{j} = 0, or equivalently, ∂f̄/∂z_{j} = 0. A holomorphic vector field on M is a complex vector field Z of type (1, 0) such that Zf is holomorphic for any locally defined holomorphic function f. If we write Z = ∑ a^{j} \frac{∂}{∂z_{j}} locally, then Z is holomorphic if and only if the components a^{j} are all holomorphic functions. Suppose that M̄ is another complex manifold, and ϕ : M → M̄ is a map. Then ϕ is said to be holomorphic if J̄ ∘ dϕ = dϕ ∘ J where J and J̄ are the complex structures of M and M̄ respectively.

If g is a Hermitian metric of M, then we have in the local coordinates

g = 2g_{jk̄}dz_{j} ⊗ dz̄_{k},  g_{jk̄} = g(Z_{j}, Zk̄)  (7.1)

because of Proposition 6.2. Thus, the Kähler form can be expressed by

ω = −2ig_{jk̄}dz_{j} ∧ dz̄_{k}.  (7.2)

By proposition 6.3, we see that g is Kähler if and only if

\frac{∂g_{jk̄}}{∂z_{l}} = \frac{∂g_{lk̄}}{∂z_{j}}  or equivalently  \frac{∂g_{jk̄}}{∂z̄_{l}} = \frac{∂g_{jl̄}}{∂z̄_{k}}.

Given an affine connection ∇ on M, we set

∇_{Z_{B}}Z_{C} := Γ_{BC}^{A}Z_{A}, A, B, C, ⋯ = 1, ⋯, n, ¯1, ⋯, n̄.

Proposition 7.1. A Hermitian metric g on M is Kähler if and only if

Γk̄ ; jl = Γ_{j̄l}^{k} = Γk̄ ; jl̄ = Γ_{j̄l̄}^{k} = 0

in each local coordinate neighborhood.

Proof. If g is Kähler, then J∇_{Z_{B}}Z_{C} = ∇_{Z_{B}}(JZ_{C}) = i∇_{Z_{B}}Z_{C} so that Γk̄ ; jl = 0 and conversely. The other relations are proved similarly. □

In the Kähler case, Γ_{BC}^{A} are determined uniquely by g as follows

g_{mk̄}Γ_{jl}^{m} = \frac{∂g_{k̄j}}{∂z_{l}} ,  g_{m̄k}Γm̄ ; j̄l̄ = \frac{∂g_{kj̄}}{∂z̄_{l}} .  (7.3)

The proof of the above fact is similar to (2.8) of Ch. 2, §2.

As in Ch. 2, §3, we can compute directly the components of the curvature tensor R_{BCD}^{A} (reps. R_{ABCD}). It has been shown in [KN] that only the components of the following type can be different from zero:

R_{jkl̄}^{h}, R_{jk̄l}^{h}, Rh̄ ; j̄kl̄, Rh̄ ; j̄k̄l,

and we have

R_{jkl̄}^{h} = −\frac{∂Γ_{jk}^{h}}{∂z̄_{l}} , R_{hj̄kl̄} = \frac{∂^{2}g_{hj̄}}{∂z_{k}∂z̄_{l}} − g^{m̄p} \frac{∂g_{hm̄}}{∂z_{k}} \frac{∂g_{j̄p}}{∂z̄_{l}} ,  (7.4)

where (g^{jk̄}) = (g_{jk̄})^{−1} so that g^{jk̄}g_{lk̄} = δ_{l}^{j}.

In the similar way (cf. Ch. 2, §3), the components R_{AB} of the Ricci tensor are as follows:

R_{jk} = Rj̄k̄ = 0,  R_{j̄k} = R̄_{jk̄,}  R_{jk̄} = − ∑_{l} \frac{∂Γ_{jl}^{l}}{∂z̄_{k}} ,  (7.5)

so that the Ricci tensor is expressed by r = 2R_{jk̄}dz_{j} ⊗ dz̄_{k}. On putting

ĝ := det(g_{jk̄}),  (7.6)

we

\frac{∂ĝ}{∂z_{k}} = ĝ ∑_{j,l} g^{jl̄}\frac{∂g_{jl̄}}{∂z_{k}},

from which together with (7.3) and (7.5) it follows that

R_{jk̄} = −\frac{∂^{2} log ĝ}{∂z_{j}∂z̄_{k}}.  (7.7)

Clearly, we have r(JX, JY) = r(X, Y) for X, Y ∈ C(T^{ℂ}M).

Definition. The Ricci form of a Kähler metric g is defined by

ρ(X, Y) := r(X, JY), ∀X, Y ∈ C(T^{ℂ}M).

In local coordinates,

ρ = −2iR_{jk̄}dz_{j} ∧ dz̄_{k}.

Proposition 7.2. The Ricci form is closed.

Proof. By (7.7), we have

ρ = 2i∂ ¯∂log ĝ,  (7.8)

which shows dρ = 0 according to (6.7). □

Therefore, The real 2-form ρ belongs to the second de Rham cohomology class of M(cf. Ch. 3, §1).

2. Holomorphic sectional curvature. Let (M, g) be a Kähler manifold, and R its Riemannian curvature tensor. Since the complex J is parallel with respect to the Levi-Civita connection ∇, then the tensor R(X, Y, Z, W) = g(R(Z, W)Y, X) (cf. Ch. 2, §2) satisfies

R(JX, JY, Z, W) = R(X, Y, JZ, JW) = R(X, Y, Z, W)  (7.9)

for X, Y, Z, W ∈ C(T^{ℂ}M).

Lemma 7.1. Let S be another (0, 4)-tensor field on M satisfying the same algebraic properties as R. If S(X, JX, X, JX) = R(X, JX, X, JX) everywhere, then S ≡ R.

Proof. We may assume that S ≡ 0; consider R − S and 0 instead of R and S. Consider the following tensor field

T (X, Y, Z, W) := R(X, JY, Z, JW) + R(X, JZ, W, JY) + R(X, JW, Y, JZ),

which vanishes for X = Y = X = W according the assumption that S = 0. Since T is symmetric in X, Y, Z and W, then T = 0 identically. Taking X = Z and Y = W, we have

2R(X, JY, X, JY) + R(X, JX, Y, JY) = 0.  (7.10)

On the other hand, by the first Bianchi identity, we have

R(X, JX, Y, JY) + R(X, Y, JY, JX) + R(X, JY, JX, Y) = 0,

which with (7.9) and other algebraic properties of R yields that

R(X, JX, Y, JY) − R(X, Y, X, Y) − R(X, Jy, X, JY) = 0.

Hence, combining with (7.10) yields that

3R(X, JY, X, JY) + R(X, Y, X, Y) = 0.  (7.11)

Replacing Y by JY in (7.11), we obtain

3R(X, Y, X, Y) + R(X, JY, X, JY) = 0.  (7.12)

From (7.11) and (7.12) it follows that

R(X, Y, X, Y) = 0,

which implies that R = 0 identically. □

For each point p ∈ M, a plane section π ⊂ T M_{p} is said to be holomorphic if π is invariant by J, i.e., Jπ = π. In such a case the sectional curvature K(π) is called the holomorphic sectional curvature by π, denoted by H(π). Taking a unit vector X ∈ π, then JX ∈ π and {X, JX} is an orthonormal basis of the holomorphic plane section π. So we have

H(π) = R(X, JX, X, JX).

From Lemma 7.1 it follows that the holomorphic sectional curvatures for all holomorphic plane sections in T M_{p} determine the curvature tensor R at p.

If the holomorphic sectional curvature is constant for all holomorphic plane sections in T M_{p} and for all points p ∈ M, then M is called a manifold with constant holomorphic sectional curvature. Adding that M is complete, it called a complex space form. It is not hard to check the following ^{[KN]}

Proposition 7.3. A Kähler manifold M has constant holomorphic sectional curvature c if and only if

\begin{aligned}R(X, Y, Z, W) & = \frac{c}{4}\{g(X, Z)g(Y, W) − g(X, W)g(Y, Z) \\ & + g(X, JZ)g(Y, JW) − g(X, JW)g(Y, JZ) + 2g(X, JY)g(Z, JW)\}; \end{aligned}

or equivalently, in complex coordinates,

R_{hj̄kl̄} = −\frac{c}{2}(g_{hj̄}g_{kl̄} + g_{hl̄}g_{j̄k}).

Clearly, the Kähler manifold with zero holomorphic sectional curvature is holomorphically isometric to ℂ^{n}.

For any positive constant c > 0, the complex projective n-space ℂP^{n} carries a Kähler metric of constant holomorphic sectional curvature c. With respect to inhomogeneous coordinates \{w_{j}\} it is given by \frac{1}{c}g, where g is expressed by (6.15).

For any negative constant c < 0, the open unit ball D_{n} := {z ∈ ℂ^{n} : ||z|| < 1} ⊂ ℂ^{n} carries a complete Kähler metric of constant holomorphic sectional curvature c. With respect to the coordinates {z_{j}} of ℂ^{n} it is given by

g = 4\frac{(1 − ∑ z_{j}z̄_{j})(∑ dz_{j} ⊗ dz̄_{j}) − (∑ z̄_{j}dz_{j}) ⊗ (∑ z_{j}dz̄_{j})}{−c(1 − ∑ z_{j}z̄_{j})^{2}},  (7.14)

called the Bergman metric.

Let π and π′ be two holomorphic planes in T M_{p} for any p ∈ M. The holomorphic bisectional curvature H(π, π′) is defined by

H(π, π′) = R(X, JX, Y, JY),  (7.15)

where X (resp. Y) is a unit vector in π (resp. π′). It is easy to verify that R(X, JX, Y, JY) depends only on π and π′. In particular, H(π, π) is the holomorphic sectional curvature. By the first Bianchi identity (Ch. 2, §3), we have

H(π, π′) = R(X, Y, X, Y) + R(X, JY, X, JY),

of which the right hand side is a sum of two sectional curvatures (up to a constant factor). We may therefore consider the concept of holomorphic bisectional curvature as an intermediate concept between those of sectional curvature and holomorphic sectional curvature.

3. Structure equations. Let M be a complex n-dimensional Kähler manifold with complex structure J and Kähler metric g. Regarding M as a real 2ndimensional Riemannian manifold, we can choose a local field of orthonormal frames {e_{1}, e¯1, ⋯, e_{n}, en̄} on T M such that ej̄ = Je_{j}. Let {ω^{A}} be the dual fields to {e_{A}}, A, B, ⋯ = 1, ⋯, n, ¯1, ⋯, n̄, so that

ωj̄ = −Jω^{j}.

Then

θ^{j} = ω^{j} + iωj̄  (7.16)

form a basis of Λ^{(1,0)}(M) which is the dual to T′M. If {ω_{B}^{A}} are the (real) connection 1-forms (cf. Ch. 3, §2), we have the structure equations (3.5) and (3.7).

On putting

θ_{k}^{j} = ω_{k}^{j} + iωj̄ ; k,  Φ_{k}^{j} = Ω_{k}^{j} + iΩj̄ ; k,  (7.17)

we easily see that (3.5) and (3.7) reduce to

\begin{matrix}dθ^{j} = −θ_{k}^{j} ∧ θ^{k}, \\ dθ_{k}^{j} = −θ_{l}^{j} ∧ θ_{k}^{l} + Φ_{k}^{j}. \end{matrix}  (7.18)

Since

ωj̄ ; k̄ = ω_{k}^{j},  ωj̄ ; k = −ωj ; k̄,  ω_{B}^{A} + ω_{A}^{B} = 0,

then we have

θ_{k}^{j} = −θ̄_{j}^{k}.

Analogously we have

Φ_{k}^{j} = −Φ̄_{j}^{k}.

Lemma 7.2. The Ricci form ρ can be expressed as

ρ = −2i ∑_{k=1}^{n} Φ_{k}^{k}.  (7.19)

Proof. Recall that R and Ω are related as follows: (cf. Ch. 3, §2)

R(X, Y)e_{A} = 2Ω_{A}^{B}(X, Y)e_{B}

for X, Y ∈ C(T M), where {e_{A}} is a local orthonormal basis on M.

On the other hand, for the Kähler manifold, one can see easily that

R(X, Y)J = JR(X, Y), R(JX, JY) = R(X, Y).

Hence, by (2.18) and the first Bianchi identity, we have

r(X, Y) = tr{Z → R(Z, X)Y }
= tr{Z → −JR(Z, X)JY }
= tr{Z → JR(X, JY)Z} + tr{Z → JR(JY, Z)X}

and

tr{Z → JR(JY, Z)X} = tr{JZ → JR(JY, JZ)X}
= tr{JZ → JR(Y, Z)X}
= tr{Z → R(Y, Z)X} = −r(Y, X) = −r(X, Y).

Therefore, we obtain

r(X, Y) = \frac{1}{2}tr\{Z → JR(X, JY)Z\} = r(JX, JY),

from which it follows that

ρ(X, Y) = r(X, JY) = −\frac{1}{2}tr\{Z → JR(X, Y)Z\}.  (7.20)

For the local field of orthonormal frames {e_{j}, ej̄ = Je_{j}}, we have

JR(X, Y)e_{j} = R(X, Y)ej̄ = 2ΩB ; j̄(X, Y)e_{B},

JR(X, Y)ej̄ = −R(X, Y)e_{j} = −2Ω_{j}^{B}(X, Y)e_{B},

from which together with (7.17) and (7.20) it follows that

\begin{aligned}ρ(X, Y) & = −\frac{1}{2} ∑_{j}\{2Ωj ; j̄(X, Y) − 2Ωj̄ ; j(X, Y)\} = 2 ∑_{j} Ωj̄ ; j(X, Y) \\ & = −2i ∑_{j}\{Ω_{j}^{j}(X, Y) + iΩj̄ ; j(X, Y)\} = −2i ∑_{j} Φ_{j}^{j}(X, Y). \end{aligned}

□

If we introduce the following matrix notations:

θ := (θ_{k}^{j}) ,  Φ := (Φ_{k}^{j}) ,  (7.21)

then (7.18)_{2} can be rewritten as

Φ = dθ − θ ∧ θ.

4. Chern classes. We consider a complex-valued function P (A_{1}, ⋯, A_{r}), whose arguments are the (n × n)-matrices A_{α}, 1 ≤ α ≤ r, and which is ℂ-linear in each of the arguments. In fact, if

A_{α} = (a_{jk}^{α}) , 1 ≤ j, k ≤ n,

then

P (A_{1}, ⋯, A_{r}) = ∑ λ_{j_{1}···j_{r}k_{1}···k_{r}}a_{j_{1}k_{1}}^{1} ⋯ a_{j_{r}k_{r}}^{r},

where λ’s are complex numbers and the summation is over the j’s and the k’s. Such a function (or polynomial) is called to be invariant if

P (uA_{1}u^{−1}, ⋯, uA_{r}u^{−1}) = P (A_{1}, ⋯, A_{r})

for any u ∈ GL(n, ℂ). It is called to be symmetric if its value remains unchanged on a permutation of its arguments.

Examples of symmetric invariant polynomials can be obtained as follows: Let A be a (n × n)-matrix, I be the unit (n × n)-matrix, and let

det (I + \frac{1}{2πi}A) = ∑_{j} ( \begin{matrix}n \\ j \end{matrix} ) P_{j}(A),  (7.22)

where P_{j}(A) is a polynomial of degree j in the elements of A. Let P_{j}(A_{1}, ⋯, A_{j}) be the completely polarized polynomial of P_{j}(A), i.e., the j-multilinear symmetric polynomial with respect to A’s, so normalized that

P_{j}(A, ⋯, A) = P_{j}(A).

From the definition (7.22), we have

P_{j}(uAu^{−1}) = P_{j}(A).

Since P_{j}(A_{1}, ⋯, A_{j}) can be expressed in terms of P_{j}(A) for different arguments A, for example,

P_{2}(A_{1}, A_{2}) = \frac{1}{2}\{P_{2}(A_{1} + A_{2}) − P_{1}(A_{1}) − P_{2}(A_{2})\},

it follows that P_{j}(A_{1}, ⋯ A_{j}) are invariant.

Definition. For a Kähler manifold M of complex dimension n, P_{j}(Φ) is called the j-th Chern class of M, denoted by c_{j}(M), where Φ is defined in (7.21).

It is easy to see that

\overline{det (I + \frac{1}{2πi}Φ)} = det (I − \frac{1}{2πi}Φ̄) = det (I + \frac{1}{2πi}Φ) ,

which implies that c_{j}(M) is a real-valued 2j-form. By the definition (7.22), we can obtain

c_{1}(M) = \frac{1}{2πi} ∑_{k} Φ_{k}^{k},  (7.23)
c_{2}(M) = −\frac{1}{8π^{2}} ∑_{j<k}(Φ_{j}^{j} ∧ Φ_{k}^{k} − Φ_{k}^{j} ∧ Φ_{j}^{k}).  (7.24)

From (7.19) and (7.23) we see that

c_{1}(M) = \frac{1}{4π} ρ.

Therefore, the first Chern class and the Ricci form for the Kähler manifolds have the same properties.

References

[BC] Bishop,R.T., Crittenden,R.J., ”Geometry of manifolds”, Academic Press, New York, 1964.

[BS] Bei,Z.G., Shen,Y.B., etc., ”Elements of Riemannian geometry” (Chinese), Higher Education Press, Beijing, 1992.

[CE] Cheeger,J., Ebin,D., ”Comparison theorems in Riemannian geometry”, North-Holland, New York, 1975.

[Cha] Chavel,I., ”Riemannian geometry: a modern introduction”, CTM 108, Cambridge, 1996.

[CH] Chern, S.S., ”Complex manifolds without potential theory”, D. VAn Nostrand Company, Inc., Princeton, 1967.

[dC] do Carmo,M., ”Riemannian geometry”, Birkhäuser, Boston, 1992.

[GH] Griffiths, P., Harris, J., ”Principles of algebraic geometry”, Wiley-Inter-science, New York, 1978.

[He] Helgason,S., ”Differential geometry, Lie groups, and symmetric spaces”, Academic Press, New York, 1978.

[KN] Kobayashi, S., Nomizu, K., ”Foundations of Differential Geometry”, Vol. II, Interscience, Wiley, New York, 1969.

[Mi] Milnor,J., ”Morse theory”, Princeton, 1963.

[SY] Schoen,R., Yau,S.T., ”Differential geometry” (Chinese), Science Press, Beijing, 1988.

[Wa] Warner,F.W., ”Foundations of differentiable manifolds and Lie groups”, GTM 94, Springer-Verlag, 1983.