Galloway's compactness theorem on Sasakian manifolds

Abstract

S. B. Myers proved in 1941 that a complete Riemannian manifold $ M^n $ with positive mean curvature is compact. W. Ambrose found in 1957 that it suffices if only the integral of the mean curvature in the tangential direction is infinite along all geodesics emanating from some points of $ M^n $ . Then G. Galloway (1981) showed that if $ V^r (r < n) $ is a compact minimal submanifold of $ M^ n $ and along each geodesic $ \gamma $ of M issuing orthogonally from V¶¶ $ lim\,inf_{l\to\infty} \int_0^l \sum_{i=1}^r K(E_i (t) \land \dot\gamma (t))dt > 0 $ ¶then M must be compact. Here K means sectional curvature, and $ E_i(t)\,i = 1,2,\dots,r $ are orthonormal vectors along $ \gamma(t) $ such that the space $ \Pi(t) $ spanned by them is parallel along $ \gamma\,to\,\Pi(0) = T_{\gamma(0)}V $ . In a Kahler manifold $ (M^{2n},J,g) $ the analogon of Galloway's theorem was obtained by K. Kenmotsu and C. Y. Xia. In their result $ V^{2r} $ is a compact immersed invariant submanifold only, which needs not be minimal. At the same time K is replaced by the sectional curvature H of M.¶In this note we prove the counterpart of Kenmotsu and Xia' theorem, i.e. the analogon of Galloway's theorem on Sasakian manifolds. We use focal points and an index form technique in a “proof to the contradiction”. In order to be able to use these tools succesfully a special vector field will be constructed.