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

Vladimir Rovenskiĭ

Annales Polonici Mathematici (1998)

  • Volume: 68, Issue: 1, page 61-82
  • ISSN: 0066-2216

Abstract

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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.

How to cite

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Vladimir Rovenskiĭ. "On the role of partial Ricci curvature in the geometry of submanifolds and foliations." Annales Polonici Mathematici 68.1 (1998): 61-82. <http://eudml.org/doc/270766>.

@article{VladimirRovenskiĭ1998,
abstract = {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.},
author = {Vladimir Rovenskiĭ},
journal = {Annales Polonici Mathematici},
keywords = {Riemannian manifold; submanifold; foliation; ruled submanifold; q-Ricci curvature; distance; radius of submanifold; index of relative nullity; -Ricci curvature},
language = {eng},
number = {1},
pages = {61-82},
title = {On the role of partial Ricci curvature in the geometry of submanifolds and foliations},
url = {http://eudml.org/doc/270766},
volume = {68},
year = {1998},
}

TY - JOUR
AU - Vladimir Rovenskiĭ
TI - On the role of partial Ricci curvature in the geometry of submanifolds and foliations
JO - Annales Polonici Mathematici
PY - 1998
VL - 68
IS - 1
SP - 61
EP - 82
AB - 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.
LA - eng
KW - Riemannian manifold; submanifold; foliation; ruled submanifold; q-Ricci curvature; distance; radius of submanifold; index of relative nullity; -Ricci curvature
UR - http://eudml.org/doc/270766
ER -

References

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