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Kaehler submanifolds of the real hyperbolic space
- Part of
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- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 66 / Issue 3 / August 2023
- Published online by Cambridge University Press:
- 22 August 2023, pp. 810-831
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- Article
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The local classification of Kaehler submanifolds
$M^{2n}$ of the hyperbolic space 
$\mathbb{H}^{2n+p}$ with low codimension 
$2\leq p\leq n-1$ under only intrinsic assumptions remains a wide open problem. The situation is quite different for submanifolds in the round sphere 
$\mathbb{S}^{2n+p}$, 
$2\leq p\leq n-1$, since Florit et al. [7] 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 in [7] for spherical submanifolds. Besides, we generalize several results obtained by Dajczer and Vlachos [5].
