Manifolds With Non-positive Sectional Curvature
In 1928, Élie Cartan proved the Cartan–Hadamard theorem: if M is a complete manifold with non-positive sectional curvature, then its universal cover is diffeomorphic to a Euclidean space. In particular, it is aspherical: the homotopy groups for i ≥ 2 are trivial. Therefore, the topological structure of a complete non-positively curved manifold is determined by its fundamental group. Preissman's theorem restricts the fundamental group of negatively curved compact manifolds.
Read more about this topic: Sectional Curvature
Famous quotes containing the word sectional:
“It is to be lamented that the principle of national has had very little nourishment in our country, and, instead, has given place to sectional or state partialities. What more promising method for remedying this defect than by uniting American women of every state and every section in a common effort for our whole country.”
—Catherine E. Beecher (1800–1878)