Riemannian Geometry - Classical Theorems in Riemannian Geometry - Geometry in Large - Sectional Curvature Bounded Above

Sectional Curvature Bounded Above

1. The Cartan–Hadamard theorem states that a complete simply connected Riemannian manifold M with nonpositive sectional curvature is diffeomorphic to the Euclidean space R^n with n = dim M via the exponential map at any point. It implies that any two points of a simply connected complete Riemannian manifold with nonpositive sectional curvature are joined by a unique geodesic.
2. The geodesic flow of any compact Riemannian manifold with negative sectional curvature is ergodic.
3. If M is a complete Riemannian manifold with sectional curvature bounded above by a strictly negative constant k then it is a CAT(k) space. Consequently, its fundamental group Γ = π₁(M) is Gromov hyperbolic. This has many implications for the structure of the fundamental group:

• it is finitely presented;
• the word problem for Γ has a positive solution;
• the group Γ has finite virtual cohomological dimension;
• it contains only finitely many conjugacy classes of elements of finite order;
• the abelian subgroups of Γ are virtually cyclic, so that it does not contain a subgroup isomorphic to Z×Z.