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Local normal forms for c-projectively equivalent metrics and proof of the Yano-Obata conjecture in arbitrary signature. Proof of the projective Lichnerowicz conjecture for Lorentzian metrics

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posted on 2015-11-20, 11:46 authored by Alexey BolsinovAlexey Bolsinov, Vladimir S. Matveev, Steffan Rosemann
Two Kähler metrics on a complex manifold are called c-projectively equivalent if their $J$-planar curves coincide. These curves are defined by the property that the acceleration is complex proportional to the velocity. We give an explicit local description of all pairs of c-projectively equivalent Kähler metrics of arbitrary signature and use this description to prove the classical Yano-Obata conjecture: we show that on a closed connected Kähler manifold of arbitrary signature, any c-projective vector field is an affine vector field unless the manifold is CPn with (a multiple of) the Fubini-Study metric. As a by-product, we prove the projective Lichnerowicz conjecture for metrics of Lorentzian signature: we show that on a closed connected Lorentzian manifold, any projective vector field is an affine vector field.

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School

  • Science

Department

  • Mathematical Sciences

Citation

BOLSINOV, A.V., MATVEEV, V.S. and ROSEMANN, S., 2015. Local normal forms for c-projectively equivalent metrics and proof of the Yano-Obata conjecture in arbitrary signature. Proof of the projective Lichnerowicz conjecture for Lorentzian metrics. arXiv:1510.00275 [math.DG]

Publisher

arXiv.org

Version

  • SMUR (Submitted Manuscript Under Review)

Publisher statement

This work is made available according to the conditions of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) licence. Full details of this licence are available at: https://creativecommons.org/licenses/by-nc-nd/4.0/

Publication date

2015

Notes

This pre-print was submitted to arXiv on 1 Oct 2015.

Language

  • en

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