Complete Einstein metrics are geodesically rigid

Abstract

We prove that every complete Einstein (Riemannian or pseudo-Riemannian) metric $g$ is geodesically rigid: if any other complete metric $\bar{g}$ has the same (unparametrized) geodesics with $g$, then the Levi-Civita connections of $g$ and $\bar{g}$ coincide.

misprints corrected, references updated. 22 pages, one .eps figure

References in corpus (3)

• A solution of a problem of Sophus Lie: Normal forms of 2-dim metrics admitting two projective vector fields
• The principle of equivalence and projective structure in space-times
• Fubini Theorem for pseudo-Riemannian metrics

Cited by in corpus (21)

• Geodesically equivalent metrics in general relativity
• There exist no 4-dimensional geodesically equivalent metrics with the same stress-energy tensor
• Einstein metrics in projective geometry
• Applications of Nijenhuis geometry II: maximal pencils of multihamiltonian structures of hydrodynamic type
• Proof of projective Lichnerowicz conjecture for pseudo-Riemannian metrics with degree of mobility greater than two
• Projectively related metrics, Weyl nullity, and metric projectively invariant equations
• Gallot-Tanno Theorem for closed incomplete pseudo-Riemannian manifolds and applications
• Projective Structure and Holonomy in 4-dimensional Lorentz Manifolds
• Holonomy and Projective Equivalence in 4-Dimensional Lorentz Manifolds
• Gallot-Tanno theorem for pseudo-Riemannian metrics and a proof that decomposable cones over closed complete pseudo-Riemannian manifolds do not exist
• The degree of mobility of Einstein metrics
• 3-dimensional Levi-Civita metrics with projective vector fields
• Optical Metrics and Projective Equivalence
• Splitting and gluing constructions for geodesically equivalent pseudo-Riemannian metrics
• Separability and Symmetry Operators for Painlevé Metrics and their Conformal Deformations
• On one class of holonomy groups in pseudo-Riemannian geometry
• Two-dimensional metrics admitting precisely one projective vector field
• Geodesically Equivalent Metrics on Homogenous Spaces
• Three Dimensional Lorentz homogeneous spaces and the Petrov classification
• Tractor calculus, BGG complexes, and the cohomology of Kleinian groups
• Quantum integrability for the Beltrami-Laplace operators of projectively equivalent metrics of arbitrary signatures