Xu, Senlin; Ni, Yilong Submanifolds of product Riemannian manifold. (English) Zbl 0979.53065 Acta Math. Sci. (Engl. Ed.) 20, No. 2, 213-218 (2000). On every product \(M=M_1\times M_2\) of Riemannian manifolds a canonical tensor field \(J_M\) is defined by \(J_M(u,v): =(u,-v)\); it satisfies \(J_M^2= \text{id}_{TM}\) and \(\nabla J_M=0\). Conversely, if on a Riemannian manifold \(M\) a tensor field \(J\) with these latter properties is given, then its eigendistributions form a parallel splitting \(TM=E_{-1} \oplus E_2\) which, by de Rham’s theorem, induces a product decomposition of \(M\) locally, and even globally if \(M\) is complete and simply connected. The authors made the following observation: If \(M\) is a complete (but not necessarily simply connected) submanifold of a Riemannian product \(N=N_1 \times N_2\) such that \(TM\) is invariant with respect to \(J_N\), then \(J_M:=J_N\mid TM\) induces a global decomposition \(M=M_1 \times M_2\) with \(M_i\subset N_i\) in this way. Furthermore, the well known representation of the second fundamental form \(h_M\) in terms of \(h_{M_1}\) and \(h_{M_2}\) is derived again and some conclusions are drawn. Reviewer: Helmut Reckziegel (Köln) Cited in 1 ReviewCited in 3 Documents MSC: 53C40 Global submanifolds 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) Keywords:Riemannian manifold; product decomposition; Riemannian product; second fundamental form PDFBibTeX XMLCite \textit{S. Xu} and \textit{Y. Ni}, Acta Math. Sci. (Engl. Ed.) 20, No. 2, 213--218 (2000; Zbl 0979.53065)