Abstract
In this paper we introduce a notion of a para-complex affine hypersphere. We give a complete local classification of such hypersurfaces and give several examples. It turns out that every para-complex affine hypersphere can be constructed from (real) affine hyperspheres. As an application, we classify all 2-dimensional para-complex affine hyperspheres.
Article PDF
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Avoid common mistakes on your manuscript.
References
Dillen, F., Vrancken, L., Verstraelen, L.: Complex affine differential geometry Atti Acc. Peloritana dei Pericolanti LXVI, pp. 232–260 (1988)
Al-Aqeel, A., Bejancu, A.: On the geometry of paracomplex submanifolds. Demonstr. Math. 34(4), 919–932 (2001)
Cruceanu, V., Fortuny, P., Gadea, P.M.: A survey on para-complex geometry. Rocky Mt. J. Math. 26(1), 83–115 (1996)
Schäfer, L.: Para-tt*-bundles on the tangent bundle of an almost para-complex manifold. Ann. Glob. Anal. Geom. 32(2), 125–145 (2007)
Lawn, M.A., Schäfer, L.: Decompositions of para-complex vector bundles and para-complex affine immersions. Results Math. 48, 246–274 (2005)
Cortés, V., Lawn, M.A., Schäfer, L.: Affine hyperspheres associated to special para-Kähler manifolds. Int. J. Geom. Methods Mod. Phys. 3, 995–1009 (2006)
Cortés, V., Mayer, C., Mohaupt, T., Saueressing, F.: Special geometry of Euclidean supersymmetry I: vector multiplets. J. High Energy Phys. 73, 3–28 (2004)
Nomizu, K., Sasaki, T.: Affine Differential Geometry. Cambridge University Press, Cambridge (1994)
Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, vol. 1. Wiley, New York (1963)
Magid, M., Vrancken, L.: Affine translation surfaces with constant sectional curvature. J. Geom. 68, 192–199 (2000)
Magid, M., Vrancken, L.: Affine translation surfaces. Results Math. 35, 134–144 (1999)
Author information
Authors and Affiliations
Corresponding author
Additional information
This Research was financed by the Ministry of Science and Higher Education of the Republic of Poland.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Szancer, Z. On Para-Complex Affine Hyperspheres. Results Math 72, 491–513 (2017). https://doi.org/10.1007/s00025-017-0676-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00025-017-0676-6