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Proof of the Projective Lichnerowicz Conjecture for Pseudo-Riemannian Metrics with Degree of Mobility Greater than Two

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Abstract

Degree of mobility of a (pseudo-Riemannian) metric is the dimension of the space of metrics geodesically equivalent to it. We prove that complete metrics on (n≥ 3)−dimensional manifolds with degree of mobility ≥ 3 do not admit complete metrics that are geodesically equivalent to them, but not affinely equivalent to them. As the main application we prove an important special 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 standard round sphere (up to a finite cover and multiplication of the metric by a constant).

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Authors and Affiliations

  1. Odessa, Ukraina

    Volodymyr Kiosak

  2. Institute of Mathematics, FSU Jena, 07737, Jena, Germany

    Vladimir S. Matveev

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  1. Volodymyr Kiosak
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  2. Vladimir S. Matveev
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Correspondence to Vladimir S. Matveev.

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Communicated by P.T. Chruściel

Partially supported by DFG (SPP 1154).

Partially supported by DFG (SPP 1154 and GK 1523).

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Kiosak, V., Matveev, V.S. Proof of the Projective Lichnerowicz Conjecture for Pseudo-Riemannian Metrics with Degree of Mobility Greater than Two. Commun. Math. Phys. 297, 401–426 (2010). https://doi.org/10.1007/s00220-010-1037-4

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