×

Submanifolds of product Riemannian manifold. (English) Zbl 0979.53065

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.

MSC:

53C40 Global submanifolds
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
PDFBibTeX XMLCite