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Critical Riemann-Finsler Metrics for the Normalized Einstein-Hilbert Functional

Yi-Bing Shen

Abstract.  In Riemannian geometry Einstein metrics are the critical point of the normalized Einstein-Hilbert functional. We generalize it to the Finsler geometry. The corresponding Euler-Lagrangian equation is derived, which depends on not only the Ricci scalar but also the mean Landsberg curvature. Some examples are given, where non-Riemannian critical metrics are related to the Cabali-Yau metrics. Moreover, some rigidity results for a Finsler metric to be Riemannian are obtained.