Proving Homeomorphism is a Diffeomorphism | Riemannian Geometry

In summary, the conversation is about defining a metric for connected smooth Riemannian manifolds and proving that a homeomorphism which is a metric isometry is also a diffeomorphism. The suggestion is to use derivatives to prove this.
  • #1
sroeyz
5
0
Hello.
Let M,N be a connected smooth riemannian manifolds.
I define the metric as usuall, the infimum of lengths of curves between the two points.
(the length is defined by the integral of the norm of the velocity vector of the curve).

Suppose phi is a homeomorphism which is a metric isometry.
I wish to prove phi is a diffeomorphism.

Please, anyone who can help.
Thanks in advance,

Roey
 
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  • #2
My instinct is to be lowbrow and just compute the derivative. Limit of ratios of distances, and all that.
 
  • #3
if you can embed them so that the metric is induced from that of euclidean space, wouldn't an isomoetry just be a restricted linearmap?

that makes it seem as if klocally it is alkways true, and derivatives are local properties.
 

1. What is the difference between a homeomorphism and a diffeomorphism?

A homeomorphism is a continuous and bijective mapping between two topological spaces, while a diffeomorphism is a smooth and bijective mapping between two differentiable manifolds. In other words, a diffeomorphism preserves not only the topological structure, but also the differentiable structure of the two spaces.

2. How do you prove that a homeomorphism is a diffeomorphism?

To prove that a homeomorphism is a diffeomorphism, we need to show that the mapping is not only continuous and bijective, but also smooth and has a smooth inverse. This can be done by checking the differentiability of the mapping and its inverse using the definition of a diffeomorphism.

3. What is the importance of proving a homeomorphism is a diffeomorphism in Riemannian geometry?

In Riemannian geometry, a diffeomorphism is used to define isometries, which are mappings that preserve the metric structure of a Riemannian manifold. This is important because isometries play a crucial role in understanding the geometric properties of a manifold, such as curvature and geodesic paths.

4. Can a homeomorphism fail to be a diffeomorphism?

Yes, a homeomorphism can fail to be a diffeomorphism. This can happen when the mapping is not smooth or does not have a smooth inverse, or when the two spaces involved have different differentiable structures. In such cases, the mapping is still a homeomorphism, but it is not considered a diffeomorphism.

5. Is it possible for a homeomorphism to be a diffeomorphism in one direction but not the other?

No, for a homeomorphism to be a diffeomorphism, it must have a smooth inverse in both directions. This means that if a mapping is a diffeomorphism in one direction, it must also be a diffeomorphism in the other direction. Otherwise, the mapping is not considered a diffeomorphism.

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