Proof of projective Lichnerowicz conjecture for pseudo-Riemannian metrics with degree of mobility greater than two

Abstract

We prove an important partial case of the pseudo-Riemannian version of the projective Lichnerowicz conjecture stating that a complete manifold admitting an essential group of projective transformations is the round sphere (up to a finite cover).

32 pages, one .eps figure. The version v1 has a misprint in Theorem 1: I forgot to write the assumption that the degree of mobility is greater than two. The versions v3, v4 have only cosmetic changes wrt v2

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• Geodesically equivalent metrics in general relativity
• There exist no 4-dimensional geodesically equivalent metrics with the same stress-energy tensor
• Gallot-Tanno Theorem for closed incomplete pseudo-Riemannian manifolds and applications
• Projectively related metrics, Weyl nullity, and metric projectively invariant equations
• Metric projective geometry, BGG detour complexes and partially massless gauge theories
• Projective Structure and Holonomy in 4-dimensional Lorentz Manifolds
• Normal forms of two-dimensional metrics admitting exactly one essential projective vector field
• Pseudo-Riemannian metrics on closed surfaces whose geodesic flows admit nontrivial integrals quadratic in momenta, and proof of the projective Obata conjecture for two-dimensional pseudo-Riemannian metrics
• The degree of mobility of Einstein metrics
• On the number of nontrivial projective transformations of closed manifolds
• 3-dimensional Levi-Civita metrics with projective vector fields
• On the groups of c-projective transformations of complete Kähler manifolds