Abstract
C∞-foliations of codimension 1 on compact Riemannian 3-manifolds are studied. New classes of foliations, namely hyperbolic, elliptic, and parabolic foliations, are considered. Examples of such foliations are presented. In particular, aC∞-metric of nonnegative sectional curvature onS 3 such that the Reeb foliation is parabolic with respect to this metric is constructed. Analytic 3-manifolds with sectional curvature of constant sign admitting parabolic foliations are classified.
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References
D. Asimov, “ Average Gaussian curvature of leaves of foliations,”Bull. Amer. Math. Soc.,84, 131–133 (1978).
F. Brito, R. Langevin, and H. Rosenberg, “Intégrales de courbure sur des variétés feuilletées,”J. Differential Geom.,16, 123–135 (1981).
Ph. Tondeur,Foliations on Riemannian Manifolds, Universitext, Springer, New York (1988).
R. Rosenberg, “Foliations by planes,”Topology,7, 131–138 (1968).
J. Cheeger and D. Gromoll, “On the structure of complete manifolds of nonnegative curvature,”Ann. of Math.,96, 413–443 (1972).
A. A. Borisenko, “On cylindrical multidimensional surfaces in Lobachevski space,”Ukrain. Geom. Sb.,33, 18–27 (1990).
Yu. A. Aminov,Geometry of Vector Fields [in Russian], Nauka, Moscow (1990).
R. Maltz, “The nullity spaces of the curvature operator,”Cahiers Topologie Géom. Différentielle Catégoriques,8, 1–20 (1968).
P. Eberlein, “Euclidean de Rham factor of a lattice of nonpositive curvature,”J. Differential Geom.,18, 209–220 (1983).
D. Johnson and L. Witt, “Totally geodesic foliations,”J. Differential Geom.,15, 225–235 (1980).
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Translated fromMatematicheskie Zametki, Vol. 63, No. 5, pp. 651–659, May, 1998.
The author wishes to express his thanks to Professor A. A. Borisenko for his supervision, and to Yu. A. Nikolaevskii for useful advice in the process of preparing the present paper.
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Bolotov, D.V. Hyperfoliations on compact 3-manifolds with restrictions on the external curvature of leaves. Math Notes 63, 575–581 (1998). https://doi.org/10.1007/BF02312836
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DOI: https://doi.org/10.1007/BF02312836