Non-immersion theorem for a class of hyperbolic manifolds
type: Item
instance of: scholarly article
title: Non-immersion theorem for a class of hyperbolic manifolds
zbMATH Open document ID: 0926.53003
DOI: 10.1016/S0926-2245(97)00017-X
author: Y. Nikolayevsky
published in: Differential Geometry and its Applications
publication date: 1999-12-02T00:00:00Z
review text: By the Cartan theorem, a hyperbolic manifold \(H^n\) of dimension \(n\) does not admit a local isometric immersion in the Euclidean space \(\mathbb{R}^{2n-2}\). Moreover, its local immersion in \(\mathbb{R}^{2n-1}\) always has a flat normal connection, that is, the first and the second fundamental forms at any point of the immersed domain of \(H^n\) are diagonal in an appropriate basis on the tangent space. Cartan's conjecture says that the hyperbolic space \(H^n\) cannot be isometrically immersed in the Euclidean space \(\mathbb{R}^{2n-1}\). This conjecture is proved under the additional hypothesis of non-simple-connectedness. The main result of the paper is following theorem. Theorem. A complete non-simply-connected hyperbolic manifold cannot be isometrically immersed in a Euclidean space with a flat normal connection. In particular, the complete hyperbolic manifold \(M^n\) with \(\pi_1(M)\neq 0\) cannot be isometrically immersed in \(\mathbb{R}^{2n-1}\). This extends some results of [\textit{F. Xavier}, Comment. Math. Helv. 60, 280-283 (1985; Zbl 0566.53045)] and is related to [\textit{Yu. A. Aminov}, Mat. Sb., Nov. Ser. 111(153), 402-433 (1980; Zbl 0431.53023)] and [\textit{J. D. Moore}, Pac. J. Math. 40, 157-166 (1972; Zbl 0238.53033)].
reviewed by: Y. Nikolayevsky
Mathematics Subject Classification ID: 53A07
Mathematics Subject Classification ID: 53C40
zbMATH DE Number: 1308133
zbMATH Keywords: isometric immersion in Euclidean space
zbMATH Keywords: hyperbolic manifold
zbMATH Keywords: flat normal connection
Wikidata QID: Q115337539
MaRDI profile type: Publication
cites work: Q3868415
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full work available at URL: s0926-2245(97)00017-x
OpenAlex ID: W2046990489
description: scientific article; zbMATH DE number 1308133