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.

117 pages; v2 added material on cones, local classification and outlook

References in corpus (5)

• Projectively related metrics, Weyl nullity, and metric projectively invariant equations
• Projective geometry and the quaternionic Feix-Kaledin construction
• Submaximally symmetric c-projective structures
• C-Projective Compactification; (quasi--)Kaehler Metrics and CR boundaries
• 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

Cited by in corpus (8)

• The Kähler geometry of Bott manifolds
• Projective geometry and the quaternionic Feix-Kaledin construction
• Submaximally symmetric c-projective structures
• C-Projective Compactification; (quasi--)Kaehler Metrics and CR boundaries
• Metrisability of projective surfaces and pseudo-holomorphic curves
• First BGG operators via homogeneous examples
• The c-projective symmetry algebras of Kähler surfaces
• Complex quaternionic manifolds and c-projective structures