Kaehler submanifolds of the real hyperbolic space

Abstract

The local classification of Kaehler submanifolds $M^{2n}$ of the hyperbolic space ${\mathbb{H}}^{2n+p}$ with low codimension $2\le p\le n-1$ under only intrinsic assumptions remains a wide open problem. The situation is quite different for submanifolds in the round sphere ${\mathbb{S}}^{2n+p}$, $2\le p\le n-1$, since Florit, Hui and Zheng have shown that the codimension has to be $p=n-1$ and then that any submanifold is just part of an extrinsic product of two-dimensional umbilical spheres in ${\mathbb{S}}^{3n-1}\subset {\mathbb{R}}^{3n}$. The main result of this paper is a version for Kaehler manifolds isometrically immersed into the hyperbolic ambient space of the result for spherical submanifolds. Besides, we generalize several results obtained by Dajczer and Vlachos.

To be published in Proceedings of the Edinburgh Mathematical Society

References in corpus (1)

• Minimal real Kaehler submanifolds