The main purpose of this paper is to develop a connectedness principle in the geometry of positive curvature. In the form, this is a surprising analog of the classical connectedness principle in complex algebraic geometry. The connectedness principle, when applied to totally geodesic immersions, provides not only a uniform formulation for the classical Synge theorem, the Frankel theorem and a recent theorem of Wilking for totally geodesic submanifolds, but also new connectedness theorems for totally geodesic immersions in the geometry of positive curvature. However, the connectedness principle may apply in certain cases which do not require the existence of totally geodesic immersions.

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Abstract. We introduce new invariants to study the asymptotic behavior of the set of rays and prove a splitting theorem for the radius of the ideal boundary of an open manifold with K ≥ 0 (Shioya's Conjecture).Mathematics Subject Classification (1991). Primary 53C20, 53C42.Keywords. Ideal boundary, set of rays. IntroductionLet M n denote an n-dimensional complete and noncompact connected riemannian manifold with secctional curvature K ≥ 0. In Theorem 2.2 in [CG] it was proved that M is diffeomorphic to the normal bundle of a totally geodesic compact submanifold S 0 , which is called the soul of M . After rigidity theorems of Strake (

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Abstract. The main purpose of this paper is to prove several connectedness theorems for complex immersions of closed manifolds in Kähler manifolds with positive holomorphic k-Ricci curvature. In particular this generalizes the classical Lefschetz hyperplane section theorem for projective varieties. As an immediate geometric application we prove that a complex immersion of an ndimensional closed manifold in a simply connected closed Kähler m-manifold M with positive holomorphic k-Ricci curvature is an embedding, provided that 2n ≥ m + k. This assertion for k = 1 follows from the Fulton-Hansen theorem (1979). IntroductionThe Lefschetz hyperplane section theorem [Le] (cf.[AF]) describes how the topology of a projective algebraic manifold X is related to the topology of a (generic) hyperplane section X 0 = X ∩ H, i.e. the relative homology groups