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Vipul at 06:10, 9 April 2007

2007-04-09T06:10:58Z

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{{Riemannian metric property}}

==Definition==

===Symbol-free definition===

A [[Riemannian manifold]] <math>(M,g)</math> (viz a [[differential manifold]] <math>M</math> equipped with a [[Riemannian metric]] <math>g</math>) is termed a '''Hadamard manifold''' if it satisfies all these three conditions:

* It is [[simply connected Riemannian manifold|simply connected]]
* It is [[complete Riemannian manifold|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.

Hadamard manifold - Revision history

Vipul: 1 revision

Vipul at 06:10, 9 April 2007