Isometric immersions of warped products
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 called a warped product immersion. The main theorem of this very nice paper is the following analogue of Moore's theorem. Let \(f: N_0 \times_{\sigma_1} N_1 \times_{\sigma_2} \dots \times_{\sigma_k} N_k\to M(c)\) be an isometric immersion into a space of constant curvature \(c\). If \(h\) is the second fundamental form of \(f\) and \(h(X_i,X_j)=0\) for all vector fields \(X_i\) and \(X_j\), tangent to \(N_i\) and \(N_j\), respectively, \(i\neq j\), then \(M\) is locally a warped product immersion. This technique, which was described in detail in the author's doctoral thesis in 1990 at the University of Köln, has already proved to be very powerful in many special situations, see for instance [\textit{F. Dillen} and \textit{S. Nölker}, J. Reine Angew. Math. 435, 33-63 (1993; Zbl 0770.53039)].
scientific article; zbMATH DE number 897463
scientific article