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PROJECTIVE STRUCTURES AND $\unicode[STIX]{x1D70C}$-CONNECTIONS

Part of: Local differential geometry Global differential geometry Classical differential geometry

Published online by Cambridge University Press:  22 March 2018

Radu Pantilie
Radu Pantilie*
Affiliation:
Institutul de Matematică “Simion Stoilow” al Academiei Române, C.P. 1-764, 014700, Bucureşti, România ([email protected])
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Abstract

We extend T. Y. Thomas’s approach to projective structures, over the complex analytic category, by involving the $\unicode[STIX]{x1D70C}$-connections. This way, a better control of projective flatness is obtained and, consequently, we have, for example, the following application: if the twistor space of a quaternionic manifold $P$ is endowed with a complex projective structure then $P$ can be locally identified, through quaternionic diffeomorphisms, with the quaternionic projective space.

Keywords

complex projective structures𝜌-connections

MSC classification

Primary: 53A20: Projective differential geometry 53B10: Projective connections 53C56: Other complex differential geometry
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 19 , Issue 2 , March 2020 , pp. 571 - 579
Copyright
© Cambridge University Press 2018

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Footnotes

The author acknowledges partial financial support from the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project no. PN-III-P4-ID-PCE-2016-0019.

References

Armstrong, S., Projective holonomy. II. Cones and complete classifications, Ann. Global Anal. Geom. 33 (2008), 137–160.Google Scholar
Atiyah, M. F., Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc. 85 (1957), 181–207.Google Scholar
Belgun, F. A., Null-geodesics in complex conformal manifolds and the LeBrun correspondence, J. Reine Angew. Math. 536 (2001), 43–63.Google Scholar
Biswas, I. and McKay, B., Holomorphic Cartan geometries and rational curves, Complex Manifolds 3 (2016), 145–168.Google Scholar
Calderbank, D. M. J., Eastwood, M. G., Matveev, V. S. and Neusser, K., C-projective geometry, Mem. Amer. Math. Soc. to appear.Google Scholar
Carrell, J. B., A remark on the Grothendieck residue map, Proc. Amer. Math. Soc. 70 (1978), 43–48.Google Scholar
Crampin, M. and Saunders, D. J., Projective connections, J. Geom. Phys. 57 (2007), 691–727.Google Scholar
Hwang, J.-M. and Mok, N., Uniruled projective manifolds with irreducible reductive G-structures, J. Reine Angew. Math. 490 (1997), 55–64.Google Scholar
Ionescu, P., Birational geometry of rationally connected manifolds via quasi-lines, in Projective Varieties with Unexpected Properties, pp. 317–335 (Walter de Gruyter GmbH & Co. KG, Berlin, 2005).Google Scholar
Kobayashi, S. and Ochiai, T., Holomorphic projective structures on compact complex surfaces, Math. Ann. 249 (1980), 75–94.Google Scholar
Kodaira, K., A theorem of completeness of characteristic systems for analytic families of compact submanifolds of complex manifolds, Ann. of Math. (2) 75 (1962), 146–162.Google Scholar
LeBrun, C. R., Spaces of complex geodesics and related structures, PhD Thesis, Oxford, 1980.Google Scholar
Mackenzie, K. C. H., Lie algebroids and Lie pseudoalgebras, Bull. London Math. Soc. 27 (1995), 97–147.Google Scholar
Molzon, R. and Mortensen, K. P., The Schwarzian derivative for maps between manifolds with complex projective connections, Trans. Amer. Math. Soc. 348 (1996), 3015–3036.Google Scholar
Pantilie, R., On the integrability of (co-)CR quaternionic manifolds, New York J. Math. 22 (2016), 1–20.Google Scholar
Pantilie, R., On the embeddings of the Riemann sphere with nonnegative normal bundles, (2013), Preprint IMAR, Bucharest, available from arXiv:1307.1993.Google Scholar
Roberts, C. W., The projective connections of T. Y. Thomas and J. H. C. Whitehead applied to invariant connections, Differ. Geom. Appl. 5 (1995), 237–255.Google Scholar
Thomas, T. Y., A projective theory of affinely connected manifolds, Math. Z. 25 (1926), 723–733.Google Scholar
Whitehead, J. H. C., The representation of projective spaces, Ann. of Math. (2) 32 (1931), 327–360.Google Scholar