Riemannian Manifolds and Completeness

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In summary, the conversation discusses how to prove that a smooth Riemannian metric on a manifold M implies that M is complete. The proposed solution involves showing that if M is totally bounded, then M is compact. One person suggests using the Hopf-Rinow theorem and the invariance of completeness under arbitrary metrics to prove this.
  • #1
Kreizhn
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Homework Statement


Suppose that for every smooth Riemannian metric on a manifold M, M is complete. Show that M is compact.

2. The attempt at a solution

I'm honestly not too sure how to start this question. If we could show that the manifold is totally bounded we would be done, but I'm not sure how to get that out of the assumption. Any ideas that I could play around with?
 
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  • #2
So I showed that M must be a closed manifold. By Hopf-Rinow if I can show it's bounded then I'll be done. I didn't use the invariance of completeness under arbitrary metrics in my closed argument so I think it will come in use for the bounded part.
 

1. What is a Riemannian manifold?

A Riemannian manifold is a mathematical space that generalizes the concept of a smooth curved surface in three dimensions. It is defined as a smooth manifold equipped with a Riemannian metric, which allows for the measurement of distances and angles on the manifold.

2. What is the significance of completeness in Riemannian manifolds?

Completeness is an important property of Riemannian manifolds that ensures that all geodesics (the shortest paths between points) can be extended infinitely without reaching a boundary or singularity. This property is crucial for many applications, such as in general relativity and optimal control theory.

3. How is completeness defined in Riemannian manifolds?

A Riemannian manifold is complete if every geodesic can be extended to infinite length without encountering any singularities or reaching a boundary. Mathematically, this means that the metric space is geodesically complete, meaning that any two points can be connected by a geodesic of finite length.

4. Can a Riemannian manifold be incomplete?

Yes, a Riemannian manifold can be incomplete if it does not satisfy the completeness property. This can occur in cases where there are singularities or boundaries that prevent the extension of geodesics. Incomplete manifolds can still be studied and have applications, but they may require special techniques to handle the incomplete parts.

5. What are some examples of Riemannian manifolds?

There are many examples of Riemannian manifolds, including the surface of a sphere, which is a two-dimensional manifold, and the three-dimensional space that we live in, which is a three-dimensional manifold. Other examples include the surface of a torus, the hyperbolic plane, and many more abstract mathematical spaces. Riemannian manifolds have applications in various fields, including physics, differential geometry, and topology.

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