Hadamard-Frankel type theorems for manifolds with partially positive curvature
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 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 3. Let \(N^{2n}\) \((n\geq 2)\) be a complete connected Kähler manifold with \(k\)-nonnegative bisectional curvature of real dimension \(2n\). Let \(W^{2r}\) and \(V^i\) be a complete immersed complex analytic submanifold of real dimension \(2r\) and a \(t(\geq 2n- r+k- 1)\)-dimensional compact immersed totally geodesic submanifold, respectively. Assume \(N^{2n}\) has \(k\)-positive bisectional curvature either at all points of \(W^{2r}\) or at all points of \(V^i\). Then \(W^{2r}\) and \(V^i\) must intersect. From Theorem 1, the authors obtain the following topological obstruction for the existence of totally geodesic submanifolds: Theorem 4. Let \(N\) be an \(n\)-dimensional complete, connected Riemannian manifold with \(k\)-positive Ricci curvature and \(V\) be an \(r\)-dimensional totally geodesic submanifold with \(2r\geq n+ k-1\), then the homomorphism of the fundamental groups: \(\pi_1 (V)\to \pi_1 (N)\) is surjective. The following compactness criterion for Kähler manifolds is also proved: Theorem 5. Let \(N\) be a complete connected Kähler manifold of complex dimension \(n\geq 2\). Let \(M\) be a compact immersed Kähler submanifold of complex dimension \(r\geq 1\). If, for any \(x\in M\) and any orthogonal basis of the form \(\{e_1, Je_1, \dots, e_r, Je_r\}\) of \(T_x M\), we have \[ \liminf_{t\to \infty} \int^i_0 \Biggl\{ \sum^r_{i=1} H(\gamma' (s), E_i (s)) \Biggr\} ds>0, \] then \(N\) is necessarily compact, where \(\gamma: [0, \infty)\to N\) is the geodesic which starts from \(x\) and is orthogonal to \(M\) at \(x\) and \(E_i\) denotes the vector field obtained by the parallel translation of \(e_i\) along \(\gamma\) for \(i=1, \dots, r\).
scientific article; zbMATH DE number 950203