Does a compact manifold always have bounded sectional curvature?

In summary, a grad student stated that a compact Riemannian manifold always has lower and upper curvature bounds. However, the person asking the question is skeptical as they have not been able to find a source explicitly stating this. The sectional curvature is defined on tangential 2-planes, which is a different manifold, and it is necessary to show that it is a continuous function on this manifold in order to prove the statement.
  • #1
Sajet
48
0
Sorry if this question seems too trivial for this forum.

A grad student at my university told me that a compact Riemannian manifold always has lower and upper curvature bounds.

Is this really true? The problem seems to be that I don't fully understand the curvature tensor's continuity etc.

What makes me a little skeptical is that I already spent quite a lot of time trying to find a source where this is explicitly stated, without any success. Usually I would expect such a statement as basic as this to appear in lots of books or lecture notes.
 
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  • #2
Sectional curvature is defined on tangential 2-planes to the manifold. It is just the Gauss curvature of each plane. So it is defined on a different manifold, the manifold of tangential 2 planes, not the original manifold. Each fiber is itself a compact manifold that is diffeomorphic to what is called the Grassmann manifold of 2 planes and the set off all of them across the entire manifold is itself a compact manifold. So you just need to convince yourself that the sectional curvature of is a continuous function on the manifold of tangential 2-planes since this manifold is compact.
 
  • #3
Thank you!

Yes, I wasn't sure whether the sectional curvature is a continuous function on the unit tangent bundle.
 

1. What is a compact manifold?

A compact manifold is a type of mathematical space that is smooth and finite in size. It can be thought of as a curved surface that can be described using mathematical equations.

2. What is sectional curvature?

Sectional curvature is a measure of how much a manifold is curved at a specific point. It takes into account the curvature in all possible directions at that point.

3. Does a compact manifold always have bounded sectional curvature?

No, a compact manifold does not always have bounded sectional curvature. It is possible for a compact manifold to have unbounded sectional curvature, meaning that the curvature can increase without limit.

4. What is the significance of bounded sectional curvature for a compact manifold?

Bounded sectional curvature is important because it ensures that the manifold is well-behaved and does not have any extreme or unpredictable curvature. This makes it easier to study and analyze the properties of the manifold.

5. Are there any exceptions to the statement that a compact manifold always has bounded sectional curvature?

Yes, there are exceptions. For example, a compact manifold with a boundary may have unbounded sectional curvature at points on the boundary. Additionally, there are non-compact manifolds that can have bounded sectional curvature.

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