C-projective geometry
Abstract
We develop in detail the theory of c-projective geometry, a natural analogue of projective differential geometry adapted to complex manifolds. We realise it as a type of parabolic geometry and describe the associated Cartan or tractor connection. A Kaehler manifold gives rise to a c-projective structure and this is one of the primary motivations for its study. The existence of two or more Kaehler metrics underlying a given c-projective structure has many ramifications, which we explore in depth. As a consequence of this analysis, we prove the Yano-Obata conjecture for complete Kaehler manifolds: if such a manifold admits a one parameter group of c-projective transformations that are not affine, then it is complex projective space, equipped with a multiple of the Fubini-Study metric.
Cite
@article{arxiv.1512.04516,
title = {C-projective geometry},
author = {David M. J. Calderbank and Michael G. Eastwood and Vladimir S. Matveev and Katharina Neusser},
journal= {arXiv preprint arXiv:1512.04516},
year = {2021}
}
Comments
117 pages; v2 added material on cones, local classification and outlook