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Twisted products in pseudo-Riemannian geometry. (English) Zbl 0792.53026

The double-twisted product \((M,g)\) of two (pseudo-) Riemannian manifolds \((M_ 1,g_ 1)\), \((M_ 2,g_ 2)\) is defined by \(M = M_ 1 \times M_ 2\) and \(g = (\lambda_ 1g_ 1) \circ \pi_ 1 + (\lambda_ 2 g_ 2) \circ \pi_ 2\dots\) where \(\lambda_ i\) are given functions on \(M\) and \(\pi_ i : M\to M_ i\) are the canonical projections, \(i = 1,2\). The twisted product is the special case \(\lambda_ 1 = 1\). The paper presents geometrical characterizations of such \((M,g)\), valid for metrics of any signatures, in terms of the canonical foliations. So \(M_ 1 \times M_ 2\) becomes a twisted product if and only if the leaves \(\{x_ 1\} \times M_ 2\) and \(M_ 1 \times \{x_ 2\}\) are mutually perpendicular and totally geodesic, resp. totally umbilic.

MSC:

53C12 Foliations (differential geometric aspects)
53C40 Global submanifolds
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
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