Isometric immersions of warped products
type: Item
instance of: scholarly article
title: Isometric immersions of warped products
zbMATH Open document ID: 0881.53052
DOI: 10.1016/0926-2245(96)00004-6
author: Stefan Nölker
published in: Differential Geometry and its Applications
publication date: 1996-08-05T00:00:00Z
review text: A well-known theorem of de Rham gives sufficient conditions for a Riemannian manifold to be (locally or globally) a Riemannian product. Also well-known is Moore's theorem that gives a sufficient condition for an isometric immersion into Euclidean space to split into a product immersion. An analogue for de Rham's theorem for warped product metrics was given by \textit{S. Hiepko} [Math. Ann. 241, 209-215 (1979; Zbl 0387.53014)]. The present paper gives an analogue of Moore's theorem for warped product immersions. Moreover, the classical notion of warped product of two Riemannian manifolds is extended to an arbitrary number of Riemannian manifolds, and an analogue of Hiepko's theorem is formulated. In particular, let \(M_0,\dots ,M_k\) be Riemannian manifolds, \(M\) their product \(M_0 \times\dots \times M_k\), and let \(\pi_i : M\to M_i\) denote the canonical projection. If \(\rho_1,\dots ,\rho_k : M_0\to \mathbb{R}_+\) are positive-valued functions, then \[ \langle X,Y\rangle:= \langle\pi_{0*}X,\pi_{0*}Y\rangle+ \sum^k_{i=1} (\rho_i \circ \pi_0)^2 \langle\pi_{i*}X, \pi_{i*}Y\rangle \] defines a Riemannian metric on \(M\), called a warped product metric, and \(M\), endowed with this metric, is denoted by \(M_0 \times_{\rho_1} M_1 \times_{\rho_2} \dots \times_{\rho_k}M_k\). First the problem of how a standard space \(M(c)\) of constant sectional curvature can be decomposed into a warped product is solved. Next warped product immersions are defined as follows. Let \(M_0 \times_{\rho_1} M_1 \times_{\rho_2} \dots \times_{\rho_k}M_k\) be a warped product and let \(f_i : N_i\to M_i\), \(i=0,\dots,k\), be isometric immersions, and define \(\sigma_i := \rho_i \circ f_0 : N_0\to \mathbb{R}_+\) for \(i = 1,\dots ,k\). Then the map \(f: N_0 \times_{\sigma_1} N_1 \times_{\sigma_2} \dots \times_{\sigma_k} N_k\to M_0 \times_{\rho_1}M_1\times_{\rho_2}\dots \times_{\rho_k} M_k\) given by \(f(p_0,\dots ,p_k) := (f_0(p_0),f_1(p_1),\dots ,f_k(p_k))\) is an isometric immersion, and is ca…
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53C40
zbMATH DE Number: 897463
zbMATH Keywords: warped products
zbMATH Keywords: warped product immersions
Wikidata QID: Q115362527
reviewed by: Franki Dillen
MaRDI profile type: Publication
full work available at URL: 0926-2245(96)00004-6
OpenAlex ID: W1970191109
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description: scientific article; zbMATH DE number 897463