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C-projective geometry
About this Title
David M. J. Calderbank, Michael G. Eastwood, Vladimir S. Matveev and Katharina Neusser
Publication: Memoirs of the American Mathematical Society
Publication Year:
2020; Volume 267, Number 1299
ISBNs: 978-1-4704-4300-9 (print); 978-1-4704-6397-7 (online)
DOI: https://doi.org/10.1090/memo/1299
Published electronically: January 4, 2021
Table of Contents
Chapters
- Introduction
- 1. Almost complex manifolds
- 2. Elements of c-projective geometry
- 3. Tractor bundles and BGG sequences
- 4. Metrisability of almost c-projective structures
- 5. Metrisability, conserved quantities and integrability
- 6. Metric c-projective structures and nullity
- 7. Global results
- 8. Outlook
Abstract
We develop in detail the theory of (almost) c-projective geometry, a natural analogue of projective differential geometry adapted to (almost) complex manifolds. We realise it as a type of parabolic geometry and describe the associated Cartan or tractor connection. A Kähler 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 Kähler 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 Kähler 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.- V. Apostolov, D. M. J. Calderbank, P. Gauduchon: The geometry of weakly selfdual Kähler surfaces, Comp. Math. 73 (2006), 359–412.
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