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Zbl 0979.53065
Xu, Senlin; Ni, Yilong
Submanifolds of product Riemannian manifold.
(English)
[J] Acta Math. Sci. (Engl. Ed.) 20, No.2, 213-218 (2000). ISSN 0252-9602

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.
[Helmut Reckziegel (Köln)]
MSC 2000:
*53C40 Submanifolds (differential geometry)
53C15 Geometric structures on manifolds

Keywords: Riemannian manifold; product decomposition; Riemannian product; second fundamental form

Cited in: Zbl 1058.53018

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