Non-immersion theorem for a class of hyperbolic manifolds
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)].
scientific article; zbMATH DE number 1308133