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arXiv · 1511.08751

Generalizing axioms of $r$-planes and $r$-spheres on Riemannian and Kähler manifolds

Cristina Levina·Sérgio Mendonça
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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 Kähler 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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Cristina Levina, Sérgio Mendonça. 2015-11-27. Generalizing axioms of $r$-planes and $r$-spheres on Riemannian and Kähler manifolds. https://arxiv.org/abs/1511.08751

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