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Generalizing axioms of $r$-planes and $r$-spheres on Riemannian and K\"ahler manifolds

Authors
  • Levina, Cristina
  • Mendonça, Sérgio
Type
Preprint
Publication Date
Nov 27, 2015
Submission Date
Nov 27, 2015
Identifiers
arXiv ID: 1511.08751
Source
arXiv
License
Yellow
External links

Abstract

The famous theorems of Cartan, related to the axiom of $r$-planes, and Leung-Nomizu about the axiom of $r$-spheres were extended to K\"ahler geometry by several authors. In this paper we replace the strong notions of totally geodesic submanifolds ($r$-planes) and extrinsic spheres ($r$-spheres) by a wider class of special isometric immersions such that theorems of type "axioms of $r$-special submanifolds" could hold. We verify also that there are plenty of special submanifolds in real and complex space forms and, in the codimension one case, in Einstein manifolds.

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