TY - JOUR

T1 - C-projective geometry

AU - Calderbank, David M. J.

AU - Eastwood, Michael G.

AU - Matveev, Vladimir S.

AU - Neusser, Katharina

N1 - Funding Information:
This article was initiated when its authors participated in a workshop at the Kioloa campus of the Australian National University in March 2013. We would like to thank the The Edith and Joy London Foundation for providing the excellent facilities at Kioloa. We would also like to thank the Group of Eight, Deutscher Akademischer Austausch Dienst, Australia-Germany Joint Research Cooperation Scheme for financially supporting the workshop in 2013 and a subsequent Kioloa workshop in 2014; for the latter, we thank, in addition, FSU Jena and the Deutsche Forschungsgemeinschaft (GK 1523/2) for their financial support. The fourth author was also supported during part of this project by Grant P201/12/G028 from the Czech Grant Agency. We would also like to thank the latter for supporting a meeting of the first, second, and fourth authors at the Mathematical Institute at Charles University in July 2014.

PY - 2020/12/31

Y1 - 2020/12/31

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85099972719&partnerID=8YFLogxK

U2 - 10.1090/memo/1299

DO - 10.1090/memo/1299

M3 - Article

VL - 267

SP - 1

EP - 150

JO - Memoirs of American Mathematical Society

JF - Memoirs of American Mathematical Society

SN - 0065-9266

IS - 1299

ER -