Intersections of Riemannian submanifolds. Variations on a theme by T.J. Frankel

Let M^n be an n-dimensional complete connected Riemannian manifold with positive sectional curvature, and let V^r and W^s be two compact, totally geodesic submanifolds of dimensions r and s. In 1961 T. Frankel proved that V r and Ws are always intersecting provided r + s ≥ n. Later a number of analogous results were achieved: for Kähler manifolds by S. I. Goldberg and S. Kobayashi, for nearly Kähler ones by A. Gray, for quaternionic Kähler manifolds by S. Marchiafava, for locally conformal Kähler manifolds by L. Ornea and for compact regular Sasakian manifolds by S. Tanno and Y. -B. Baik. Recently K. Kenmotsu and C. Xia, have obtained similar results on manifolds with partially positive curvature. In this paper we prove results of similar type in a Riemannian space for two minimal hypersurfaces. We present a construction of Sasakian manifolds with k-positive bisectional curvature. In analogy to the Kähler case, an orthonormal vector system is obtained in a Sasakian manifold, which is parallel along a geodesic. Using this vector system, we prove an intersection theorem of Frankel type for two compact invariant submanifolds of a Sasakian space with k-positive bisectional curvature.