Myers-Bonnet theorem: Difference between revisions
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===Cartan-Hadamard theorem=== | ===Cartan-Hadamard theorem=== | ||
{{further|Cartan-Hadamard theorem]]}} | |||
The Cartan-Hadamard theorem talks of the analogous statement when the manifold has negative curvature throughout. It says that under that assumption, the universal cover is diffeomorphic to real Euclidean space. | The Cartan-Hadamard theorem talks of the analogous statement when the manifold has negative curvature throughout. It says that under that assumption, the universal cover is diffeomorphic to real Euclidean space. | ||
Together, the Cartan-Hadamard theorem and Myers-Bott theorem tell us that a manifold which has positive curvature bounded from below, cannot be diffeomorphic to a manifold which has negative sectional curvature throughout. | Together, the Cartan-Hadamard theorem and Myers-Bott theorem tell us that a manifold which has positive curvature bounded from below, cannot be diffeomorphic to a manifold which has negative sectional curvature throughout. | ||
Revision as of 13:10, 18 June 2007
This article describes a result related to the sectional curvature of a Riemannian manifold
This result relates information on curvature to information on topology of a manifold
This article makes a prediction about the universal cover of a manifold based on given data at the level of a:topological manifold
This result is valid in all dimensions
Statement
If a Riemannian manifold has the property that there exists a positive constant that lower-bounds the sectional curvature for all tangent planes at all points, then the manifold is compact with finite fundamental group. This is equivalent to saying that the universal cover of the manifold is compact.
Relation with other results
Cartan-Hadamard theorem
Further information: Cartan-Hadamard theorem]] The Cartan-Hadamard theorem talks of the analogous statement when the manifold has negative curvature throughout. It says that under that assumption, the universal cover is diffeomorphic to real Euclidean space.
Together, the Cartan-Hadamard theorem and Myers-Bott theorem tell us that a manifold which has positive curvature bounded from below, cannot be diffeomorphic to a manifold which has negative sectional curvature throughout.