Hadamard manifold

This article defines a property that makes sense for a Riemannian metric over a differential manifold

Definition

Symbol-free definition

A Riemannian manifold $(M,g)$ (viz a differential manifold $M$ equipped with a Riemannian metric $g$) is termed a Hadamard manifold if it satisfies all these three conditions:

• It is simply connected
• It is complete
• The sectional curvature is everywhere nonpositive

Facts

• The universal Riemannian covering space for any complete Riemannian manifold of everywhere nonpositive sectional curvature is a Hadamard manifold.
• On account of its being simply connected, we can talk of the uniquegeodesic joining any two points.