Hadamard manifold

In differential geometry , a branch of mathematics , Hadamard manifolds are simply connected complete Riemannian manifolds with non-positive sectional curvature.

definition

A Hadamard manifold is a simply connected complete Riemann manifold of non-positive section curvature .

properties

Hadamard manifolds are CAT (0) -spaces - this follows from Toponogow's theorem .

Hadamard manifolds are contractible - this follows from the Cartan-Hadamard theorem .

Examples

• the Euclidean space ${\displaystyle \mathbb {R} ^{n}}$
• the hyperbolic space
• ${\displaystyle SL(n,\mathbb {R} )/SO(n)}$
• more generally all symmetrical spaces without a compact factor
• Products of Hadamard Manifolds

literature

• Patrick Barry Eberlein, Barrett O'Neill: Visibility manifolds. In: Pacific Journal of Mathematics. Vol. 46, No. 1, 1973, ISSN  0030-8730 , 45-109, doi : 10.2140 / pjm.1973.46.45 .
• Werner Ballmann , Mikhael Gromov , Viktor Schroeder: Manifolds of nonpositive curvature (= Progress in Mathematics. 61). Birkhäuser, Boston et al. 1985, ISBN 3-7643-3181-X .