Splitting theorem: Difference between revisions
No edit summary |
m (3 revisions) |
||
| (2 intermediate revisions by the same user not shown) | |||
| Line 3: | Line 3: | ||
==Statement== | ==Statement== | ||
Any [[geodesically complete Riemannian manifold|complete]] Riemannian manifold whose [[Ricci curvature]] is everywhere nonnegative can be expressed as a direct product of a [[ | Any [[geodesically complete Riemannian manifold|complete]] Riemannian manifold whose [[Ricci curvature]] is everywhere nonnegative can be expressed as a direct product of a [[Euclidean space]] and a Riemannian manifold which does not contain any [[line]]. (A line here is a geodesic every finite segment of which realizes the distance between its endpoints). | ||
==Relation with other results== | ==Relation with other results== | ||
| Line 11: | Line 11: | ||
{{further|[[Bonnet-Myers theorem]]}} | {{further|[[Bonnet-Myers theorem]]}} | ||
The Bonnet-Myers theorem states a very similar result for a manifold which has positive Ricci curvature everywhere, bounded below by a positive constant. Note that the Bonnet-Myers theorem, which asserts that any such manifold is compact, implies that it does not contain any line. Thus, the splitting theorem is closely related to the Bonnet-Myers theorem, | The Bonnet-Myers theorem states a very similar result for a manifold which has positive Ricci curvature everywhere, bounded below by a positive constant. Note that the Bonnet-Myers theorem, which asserts that any such manifold is compact, implies that it does not contain any line. Thus, the splitting theorem is closely related to the Bonnet-Myers theorem, the difference being that it allows zero curvature. | ||
Latest revision as of 20:09, 18 May 2008
This article describes a result related to the Ricci curvature of a Riemannian manifold
Statement
Any complete Riemannian manifold whose Ricci curvature is everywhere nonnegative can be expressed as a direct product of a Euclidean space and a Riemannian manifold which does not contain any line. (A line here is a geodesic every finite segment of which realizes the distance between its endpoints).
Relation with other results
Bonnet-Myers theorem
Further information: Bonnet-Myers theorem
The Bonnet-Myers theorem states a very similar result for a manifold which has positive Ricci curvature everywhere, bounded below by a positive constant. Note that the Bonnet-Myers theorem, which asserts that any such manifold is compact, implies that it does not contain any line. Thus, the splitting theorem is closely related to the Bonnet-Myers theorem, the difference being that it allows zero curvature.