Abstract. We investigate the topological structure of Alexandrov surfaces of curvature bounded below which possess convex functions. We do not assume the continuities of these functions. Nevertheless, if the convex functions satisfy a condition of local nonconstancy, then the topological structures of Alexandrov surfaces and the level sets configurations of these functions in question are determined.
IntroductionTypical examples of convex functions on Riemannian manifolds are Busemann functions on complete Riemannian manifolds of nonnegative sectional curvature. Using filtration by compact totally convex sets, J. Cheeger and D. Gromoll investigated the structure theorem by which the topological structure of complete noncompact Riemannian manifolds of nonnegative sectional curvature is determined ([5]).It is well known that the existence of a locally nonconstant convex function on a complete Riemannian manifold M imposes strong restrictions on the topology of M ([6], Theorems C, D and F). Here, the local nonconstancy of a convex function means that it is not constant on any open set. As is explained on p.130 of [6], every noncompact smooth manifold M admits a complete Riemannian metric g and a nontrivial smooth function ϕ such that ϕ is convex with respect to g. For such a ϕ, the minimum set of ϕ has a nonempty interior in which all the topological information of M is included. Thus it is natural to assume that a convex function is locally nonconstant. Geodesic completeness on M plays an essential role in the above results. In fact, the property of local Lipschitz continuity of a convex function on M is derived from geodesic completeness, and also the noncompactness of M .It is natural to ask if certain topological restrictions can be obtained for geodesic spaces admitting locally nonconstant convex functions. In fact, N. Innami proved in Theorem 3.13 of [9] that if a Busemann G-surface admits a locally nonconstant convex function, then it is homeomorphic to either a plane R 2 , a cylinder S 1 × R 1 , or an open Möbius strip. Here the geodesic completeness holds on Busemann Gsurfaces.
We prove that the asymptotic cone of every complete, connected, non-compact Riemannian manifold of roughly non-negative radial curvature exists, and it is isometric to the Euclidean cone over their Tits ideal boundaries.Obviously, the Gaussian curvature G( p) at p ∈ M is equal to K d M ( õ, p) . We say that the radial sectional curvature at o of M is bounded below by
We discuss codimension one isometric immersions of complete Riemannian manifolds into the projective spaces with constant holomorphic sectional curvature. Here, the shape operator and the curvature transformation with respect to the normal unit have the same eigenspaces. We then characterize the metric spheres in terms of the shape operator.
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