Abstract
In Riemannian geometry and its applications, the most popular is the class of Riemannian submersions (and foliations) [1–4] which are characterized by simplest mutual disposition of fibers. The purpose of the present article is to introduce other, more general, classes of submersions of Riemannian manifolds which, as well as the class of Riemannian submersions, are described by simple local properties of configuration tensors and to begin their study.
Given a submersion π:M→M of differentiable manifolds with compact connected fibers and any metric onM, we define a metric on the base with the help of theL 2-norm of horizontal fields. In this caseT¯ M becomes a subbundle of some larger bundleM. The main class of totally geodesic submersions introduced in the article (Definition 1) corresponds to the metrics onM with simplest disposition ofT¯ M inM. In the article we obtain a criterion for such submersions (Corollary 1); existence is proved by means of the product with a metric varying along fibers (Example 2). To study totally geodesic submersions, we use ideas from the theory of Riemannian submersions and submanifolds with degenerate second form (Theorems 1 and 2 and Corollary 4).
Foliations modeled by totally geodesic submersions (see equality (13)) are of interest too, but we leave them beyond the scope of the article.
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References
B. O'Neill, “The fundamental equations of submersion,” Michigan Math. J.,13, 459–469 (1966).
A. Gray, “Pseudo-Riemannian almost product manifolds and submersions,” J. Math. Mech.,16, 715–737 (1967).
A. L. Besse, Einstein Manifolds. Vol. II [Russian translation], Mir, Moscow (1990).
A. Borisenko and A. Yampol'skiî, “Riemannian geometry of bundles,” Uspekhi Mat. Nauk,46, No. 6, 51–95 (1991).
C. Ehresmann, “Les connexions infinitésimales dans un expace fibré différentiable,” Colloque de Topologie, Bruxelles, 29–55 (1950).
S. Stepanov, “Weil submersions,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 5, 93–95 (1992).
V. Rovenskiî, “Twisted Hopf bundles,” Ukrain. Geom. Sb., No. 33, 101–114 (1990).
A. L. Besse, Manifolds All Whose Geodesics Are Closed [Russian translation], Mir, Moscow (1981).
R. Maltz, “The nullity spaces of curvative-like tensors,” J. Differential Geom.,7, 519–525 (1972).
Yu. Lumiste, “Semisymmetric submanifolds,” in: Geometry. Vol. 23 (Itogi Nauki i Tekhniki) [in Russian], VINITI, Moscow, 1991.
S. Tanno, “Totally geodesic foliations with compact leaves,” Hokkaido Math. J.,1, 7–11 (1972).
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This work was supported by the Russian Foundation for Fundamental Research (Grant 94-01-00271).
Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 35, No. 5, pp. 1154–1164, September–October, 1994.
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Rovenskiî, V.Y. Classes of submersions of Riemannian manifolds with compact fibers. Sib Math J 35, 1027–1035 (1994). https://doi.org/10.1007/BF02104581
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DOI: https://doi.org/10.1007/BF02104581