# Hadamard manifold

From Diffgeom

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

## Definition

### Symbol-free definition

A Riemannian manifold (viz a differential manifold equipped with a Riemannian metric ) 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.