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Annales Polonici Mathematici

1998 | 68 | 1 | 61-82
Tytuł artykułu

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

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Słowa kluczowe
EN
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
61-82
Opis fizyczny
Daty
wydano
1998
otrzymano
1996-11-28
Twórcy
autor
• Geometry Chair, State Pedagogical University, Lebedeva St., 89 Krasnoyarsk 660049, Russia
Bibliografia
• [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.
• [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.
• [Bor 1] A. Borisenko, Complete l-dimensional submanifolds of nonpositive extrinsic curvature in a Riemannian space, Mat. Sb. 104 (1977), 559-576 (in Russian).
• [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).
• [Bor 3] A. Borisenko, On extremal properties of compact parabolic submanifolds in a Riemannian space, Mat. Sb. 133 (1987), 112-126 (in Russian).
• [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.
• [BRT] A. Borisenko, M. Rabelo and K. Tenenblat, On saddle submanifolds of Riemannian manifolds, in: Conf. on Diff. Geometry, Abstracts, Budapest, 1996, 24-25.
• [DR] M. Dajczer and L. Rodríguez, On isometric immersions into complex space forms, Math. Ann. 299 (1994), 223-230.
• [Fer] D. Ferus, Totally geodesic foliations, Math. Ann. 188 (1970), 313-316.
• [Flo] L. Florit, On submanifolds with nonpositive extrinsic curvature, Math. Ann. 298 (1994), 187-192.
• [Fra] T. Frankel, Manifolds with positive curvature, Pacific J. Math. 11 (1961), 165-171.
• [Gla] V. Glazyrin, Topological and metric properties of k-saddle submanifolds, Dokl. Akad. Nauk SSSR 233 (1977), 1028-1030 (in Russian).
• [GK] S. Goldberg and S. Kobayashi, On holomorphic bisectional curvature, J. Differential Geom. 1 (1967), 225-233.
• [Ish] T. Ishihara, Radii of immersed manifolds and nonexistence of immersions, Proc. Amer. Math. Soc. 78 (1980), 276-279.
• [Jac] M. Jacobowitz, Isometric embedding of a compact Riemannian manifold into Euclidean space, Proc. Amer. Math. Soc. 40 (1973), 245-246.
• [KX 1] K. Kenmotsu and C. Xia, Hadamard-Frankel type theorems for manifolds with partially positive curvature, Pacific J. Math., to appear.
• [KX 2] K. Kenmotsu and C. Xia, Intersections of minimal submanifolds in manifolds of partially positive curvature, Kodai Math. J. 18 (1995), 242-249.
• [KN] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vols. 1, 2, Interscience Publ., 1963, 1969.
• [Mal] R. Maltz, The nullity spaces of curvature-like tensors, J. Differential Geom. 7 (1972), 519-525.
• [Moo 1] J. Moore, An application of second variation to submanifold theory, Duke Math. J. 42 (1975), 191-193.
• [Moo 2] J. Moore, Submanifolds of constant positive curvature, I, Duke Math. J. 44 (1977), 449-484.
• [Mor] J. Morvan, Distance of two submanifolds of a manifold with positive curvature, Rend. Mat. 3 (1983), 357-366.
• [O'N] B. O'Neill, Immersion of manifolds of nonpositive curvature, Proc. Amer. Math. Soc. 11 (1960), 132-134.
• [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.
• [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).
• [Shef] S. Shefel', On two classes of k-dimensional submanifolds in n-dimensional Euclidean space, Sibirsk. Mat. Zh. 10 (1969), 459-467 (in Russian).
• [Shen] Z. Shen, On complete manifolds of nonnegative kth-Ricci curvature, Trans. Amer. Math. Soc. 338 (1993), 289-310.
• [Tan] S. Tanno, Totally geodesic foliations with compact leaves, Hokkaido Math. J. 1 (1972), 7-11.
• [Top] V. Toponogov, Extremal theorems for Riemannian spaces with curvature bounded above. I, Sibirsk. Mat. Zh. 15 (1974), 1348-1371 (in Russian).
• [Wu] H. Wu, Manifolds of partially positive curvature, Indiana Univ. Math. J. 36 (1987), 525-548.
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Bibliografia
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