×

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
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bishop, R. L., ?Clairaut submersions?,Differential Geometry, in Honor of K. Yano, Kinokuniya, Tokyo, 1972, pp. 21-31. · Zbl 0246.53048
[2] Bishop, R. L. and O’Neill, B., ?Manifolds of negative curvature?,Trans. Amer. Math. Soc. 145 (1969), 1-49. · Zbl 0191.52002 · doi:10.1090/S0002-9947-1969-0251664-4
[3] Blumenthal, R. A. and Hebda, J. J., ?De Rham decomposition theorems for a foliated manifolds?,Ann. Inst. Fourier 33 (1983), pp. 183-198. · Zbl 0487.57010
[4] Blumenthal, R. A. and Hebda, J. J., ?An analogue of the holonomy bundle for a foliated manifold?,Tôhoku Math. J. 40 (1988), pp. 189-197. · Zbl 0632.53033 · doi:10.2748/tmj/1178228025
[5] Chen, B.-Y.,Geometry of Submanifolds, Marcel Dekker, New York, 1973. · Zbl 0262.53036
[6] Chen, B.-Y.,Geometry of Submanifolds and its Applications, Science University of Tokyo, 1981. · Zbl 0474.53050
[7] Ehresmann, C., ?Les connexions infinitésimales dans un espace fibré différentiable?,Colloque de Topologie, Bruxelles, 1950.
[8] Escobales, R. Jr and Parker, P. E., ?Geometric consequences of the normal curvature cohomology class in umbilic foliations?,Indiana Univ. Math. J. 37 (1988), pp. 389-408. · Zbl 0665.53035 · doi:10.1512/iumj.1988.37.37020
[9] Hicks, N., ?A theorem on affine connexions?,Illinois J. Math. 3 (1959), 242-254. · Zbl 0088.38103
[10] Hermann, R., ?A sufficient condition that a mapping of Riemannian manifolds be a fibre bundle?,Proc. Amer. Math. Soc. 11 (1960), pp. 236-242. · Zbl 0112.13701 · doi:10.1090/S0002-9939-1960-0112151-4
[11] Hiepko, S., ?Eine innere Kennzeichnung der verzerrten Produkte?,Math. Ann. 241 (1979), pp. 209-215. · Zbl 0395.53021 · doi:10.1007/BF01421206
[12] Hiepko, S. and Reckziegel, H., ?Über sphärische Blätterungen und die Vollständigkeit ihrer Blätter?,Manuscripta Math. 31 (1980), pp. 269-283. · Zbl 0441.53035 · doi:10.1007/BF01303277
[13] Kobayashi, S. and Nomizu, K.,Foundations of Differential Geometry, Vol. I, Interscience Publishers, New York, 1963. · Zbl 0119.37502
[14] O’Neill, B.,Semi-Riemannian Geometry, Academic Press, New York, 1983.
[15] Poor, W. A.,Differential Geometric Structures. McGraw-Hill, New York, 1981. · Zbl 0493.53027
[16] Wu, H., ?On the de Rham decomposition theorem?,Illinois J. Math. 8 (1964), 291-311. · Zbl 0122.40005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.