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

TL;DR

This paper provides explicit local descriptions of c-projectively equivalent Kähler metrics of any signature and proves the Yano-Obata and Lichnerowicz conjectures, establishing conditions under which projective and c-projective vector fields are affine.

Contribution

It introduces a local normal form for c-projectively equivalent Kähler metrics and proves the Yano-Obata and Lichnerowicz conjectures in arbitrary signature, including Lorentzian cases.

Findings

C-projectively equivalent Kähler metrics are characterized explicitly.

On closed Kähler manifolds, c-projective vector fields are affine unless the manifold is complex projective space.

On closed Lorentzian manifolds, all projective vector fields are affine.

Abstract

Two K\"ahler 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 K\"ahler metrics of arbitrary signature and use this description to prove the classical Yano-Obata conjecture: we show that on a closed connected K\"ahler manifold of arbitrary signature, any c-projective vector field is an affine vector field unless the manifold is $CP^n$ 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.