Hadamard-Frankel type theorems for manifolds with partially positive curvature
type: Item
instance of: scholarly article
title: Hadamard-Frankel type theorems for manifolds with partially positive curvature
zbMATH Open document ID: 0865.53053
author: Katsuei Kenmotsu
author: Changyu Xia
published in: Pacific Journal of Mathematics
publication date: 1996-12-15T00:00:00Z
review text: A well-known theorem of Frankel states that if \(N\) is an \(n\)-dimensional complete connected Riemannian manifold with strictly positive sectional curvature and if \(V\) is an \(r\)-dimensional compact totally geodesic immersed submanifold of \(N\) with \(2r\geq n\), then the homomorphism of the fundamental groups: \(\pi_1 (V)\to \pi_1 (N)\) is surjective. Unfortunately, the set of manifolds with positive sectional curvature is not so big. Frankel also showed that if \(N\) is a complete connected Kähler manifold with positive sectional curvature, then any two compact analytic submanifolds must intersect if their dimension sum is at least that of \(N\). Since a complete connected Kähler manifold of positive bisectional curvature which contains a compact Kähler submanifold is compact (see Theorem 5 below), the above \(N\) is actually compact, thus it is biholomorphic to a complex projective space by the settlement of Frankel's conjecture. The topology of \(N\) is therefore very simple. The authors consider the situation where \(N\) can only have either partially positive sectional curvature or partially positive bisectional curvature. Such manifolds cover many known examples. Firstly, the authors generalize Frankel's theorems to this class of manifolds: Theorem 1. Let \(N\) be an \(n\)-dimensional complete connected Riemannian manifold with \(k\)-nonnegative Ricci curvature and let \(V\) and \(W\) be two compact immersed totally geodesic submanifolds of dimensions \(r\) and \(s\), respectively. Assume \(N\) has \(k\)-positive Ricci curvature either at all points of \(V\) or at all points of \(W\). If \(r+s\geq n+k -1\), then \(V\) and \(W\) must intersect. Theorem 2. Let \(N\) be a complete connected Kähler manifold with \(k\)-nonnegative bisectional curvature of complex dimension \(n\). Let \(V\) and \(W\) be two compact complex analytic submanifolds in \(N\) of complex dimension \(r\) and \(s\), respectively. Assume \(N\) has \(k\)-positive bisectional curvature e…
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53C55
zbMATH DE Number: 950203
zbMATH Keywords: partially positive curvature
zbMATH Keywords: nonnegative Ricci curvature
zbMATH Keywords: intersection of submanifolds
zbMATH Keywords: nonnegative bisectional curvature
zbMATH Keywords: totally geodesic submanifolds
zbMATH Keywords: Kähler manifold
MaRDI profile type: Publication
full work available at URL: pjm.1996.176.129
OpenAlex ID: W1499391941
DOI: 10.2140/PJM.1996.176.129
description: scientific article; zbMATH DE number 950203