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

Vladimir Rovenskiĭ¹

¹

Annales Polonici Mathematici, Tome 68 (1998) no. 1, pp. 61-82.

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

DOI : 10.4064/ap-68-1-61-82

Keywords: Riemannian manifold, submanifold, foliation, ruled submanifold, q-Ricci curvature, distance, radius of submanifold, index of relative nullity