On the role of partial Ricci curvature in the geometry of submanifolds and foliations

Vladimir Rovenskiĭ

Annales Polonici Mathematici, Tome 69 (1998), p. 61-82 / Harvested from The Polish Digital Mathematics Library

Résumé

Submanifolds and foliations with restrictions on q-Ricci curvature are studied. In §1 we estimate the distance between two compact submanifolds in a space of positive q-Ricci curvature, and give applications to special classes of submanifolds and foliations: k-saddle, totally geodesic, with nonpositive extrinsic q-Ricci curvature. In §2 we generalize a lemma by T. Otsuki on asymptotic vectors of a bilinear form and then estimate from below the radius of an immersed submanifold in a simply connected Riemannian space with nonpositive curvature; moreover, we prove a theorem on nonembedding into a circular cylinder when the ambient space is Euclidean. Corollaries are nonembedding theorems of Riemannian manifolds with nonpositive q-Ricci curvature into a Euclidean space. In §3 a lower estimate of the index of relative nullity of a submanifold with nonpositive extrinsic q-Ricci curvature is proven. Corollaries are extremal theorems for a compact submanifold with the nullity foliation in a Riemannian space of positive curvature. On the way, some results by T. Frankel, K. Kenmotsu and C. Xia, J. Morvan, A. Borisenko, S. Tanno, B. O'Neill, J. Moore, T. Ishihara, H. Jacobowitz, L. Florit, M. Dajczer and L. Rodríguez are generalized.

Détail

Publié le : 1998-01-01

Zbl 0910.53019

EUDML-ID : urn:eudml:doc:270766

Bibliographie

[000] [Abe 1] K. Abe, A characterization of totally geodesic submanifolds in $S^N$ and $CP^N$ by an inequality, Tôhoku Math. J. 23 (1971), 219-244. | Zbl 0245.53053

[001] [Abe 2] K. Abe, Applications of a Riccati type differential equation to Riemannian manifolds with totally geodesic distributions, Tôhoku Math. J. 25 (1973), 425-444. | Zbl 0283.53045

[002] [Bor 1] A. Borisenko, Complete l-dimensional submanifolds of nonpositive extrinsic curvature in a Riemannian space, Mat. Sb. 104 (1977), 559-576 (in Russian). | Zbl 0372.53029

[003] [Bor 2] A. Borisenko, On external geometrical properties of parabolic submanifolds and topological properties of saddle submanifolds in symmetric spaces of rank one, Mat. Sb. 116 (1981), 440-457 (in Russian). | Zbl 0477.53051

[004] [Bor 3] A. Borisenko, On extremal properties of compact parabolic submanifolds in a Riemannian space, Mat. Sb. 133 (1987), 112-126 (in Russian). | Zbl 0628.53049

[005] [Bor 4] A. Borisenko, The foliations of extrinsic negative curvature in a Riemannian space, in: Conf. on Diff. Geometry and Applications, Abstracts, Brno, 1995, 5-6.

[006] [BRT] A. Borisenko, M. Rabelo and K. Tenenblat, On saddle submanifolds of Riemannian manifolds, in: Conf. on Diff. Geometry, Abstracts, Budapest, 1996, 24-25.

[007] [DR] M. Dajczer and L. Rodríguez, On isometric immersions into complex space forms, Math. Ann. 299 (1994), 223-230. | Zbl 0806.53019

[008] [Fer] D. Ferus, Totally geodesic foliations, Math. Ann. 188 (1970), 313-316. | Zbl 0194.52804

[009] [Flo] L. Florit, On submanifolds with nonpositive extrinsic curvature, Math. Ann. 298 (1994), 187-192. | Zbl 0810.53011

[010] [Fra] T. Frankel, Manifolds with positive curvature, Pacific J. Math. 11 (1961), 165-171. | Zbl 0107.39002

[011] [Gla] V. Glazyrin, Topological and metric properties of k-saddle submanifolds, Dokl. Akad. Nauk SSSR 233 (1977), 1028-1030 (in Russian).

[012] [GK] S. Goldberg and S. Kobayashi, On holomorphic bisectional curvature, J. Differential Geom. 1 (1967), 225-233. | Zbl 0169.53202

[013] [Ish] T. Ishihara, Radii of immersed manifolds and nonexistence of immersions, Proc. Amer. Math. Soc. 78 (1980), 276-279. | Zbl 0438.53054

[014] [Jac] M. Jacobowitz, Isometric embedding of a compact Riemannian manifold into Euclidean space, Proc. Amer. Math. Soc. 40 (1973), 245-246. | Zbl 0265.53047

[015] [KX 1] K. Kenmotsu and C. Xia, Hadamard-Frankel type theorems for manifolds with partially positive curvature, Pacific J. Math., to appear. | Zbl 0865.53053

[016] [KX 2] K. Kenmotsu and C. Xia, Intersections of minimal submanifolds in manifolds of partially positive curvature, Kodai Math. J. 18 (1995), 242-249.

[017] [KN] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vols. 1, 2, Interscience Publ., 1963, 1969. | Zbl 0119.37502

[018] [Mal] R. Maltz, The nullity spaces of curvature-like tensors, J. Differential Geom. 7 (1972), 519-525. | Zbl 0272.53015

[019] [Moo 1] J. Moore, An application of second variation to submanifold theory, Duke Math. J. 42 (1975), 191-193. | Zbl 0337.53045

[020] [Moo 2] J. Moore, Submanifolds of constant positive curvature, I, Duke Math. J. 44 (1977), 449-484. | Zbl 0361.53050

[021] [Mor] J. Morvan, Distance of two submanifolds of a manifold with positive curvature, Rend. Mat. 3 (1983), 357-366. | Zbl 0533.53049

[022] [O'N] B. O'Neill, Immersion of manifolds of nonpositive curvature, Proc. Amer. Math. Soc. 11 (1960), 132-134. | Zbl 0123.38604

[023] [Ots] T. Otsuki, On the existence of solutions of a system of quadratic equations and its geometrical application, Proc. Japan Acad. 29 (1953), 99-100. | Zbl 0052.17602

[024] [Rov] V. Rovenskiĭ, Submanifolds and foliations with restrictions on partial Ricci curvature, in: Problems of Mathematical Analysis, Krasnoyarsk Technical Univ., 1996, 53-62 (in Russian).

[025] [Shef] S. Shefel', On two classes of k-dimensional submanifolds in n-dimensional Euclidean space, Sibirsk. Mat. Zh. 10 (1969), 459-467 (in Russian).

[026] [Shen] Z. Shen, On complete manifolds of nonnegative kth-Ricci curvature, Trans. Amer. Math. Soc. 338 (1993), 289-310. | Zbl 0783.53026

[027] [Tan] S. Tanno, Totally geodesic foliations with compact leaves, Hokkaido Math. J. 1 (1972), 7-11. | Zbl 0251.53034

[028] [Top] V. Toponogov, Extremal theorems for Riemannian spaces with curvature bounded above. I, Sibirsk. Mat. Zh. 15 (1974), 1348-1371 (in Russian).

[029] [Wu] H. Wu, Manifolds of partially positive curvature, Indiana Univ. Math. J. 36 (1987), 525-548. | Zbl 0639.53050