Diffeomorphisms and embedded submanifolds

In summary, a diffeomorphism is a smooth, bijective function between two differentiable manifolds that has a smooth inverse. An embedded submanifold is a subset of a larger manifold that is itself a manifold. Diffeomorphisms are often used to transform or map one manifold onto another, and this can be used to show that two manifolds are diffeomorphic. Embedded submanifolds are often defined using diffeomorphisms, and have applications in differential geometry, physics, computer graphics, and computer vision. However, using diffeomorphisms and embedded submanifolds can be challenging due to difficulty in visualization and manipulation, as well as the complexity of finding a diffeomorphism between
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osprey
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The tex-code in my post was behaving really weirdly - some parts of the code seemed to have "exchanged places", so I chose to delete what I had written. I apologize for the inconvenience...
 
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It does that sometimes when using Preview.
If you post it and hit Refresh a couple of times, it usually shows up correctly :)
 

1. What is a diffeomorphism?

A diffeomorphism is a smooth, bijective function between two differentiable manifolds that has a smooth inverse. It essentially means that the two manifolds are "diffeomorphic" or "smoothly equivalent" to each other.

2. What is an embedded submanifold?

An embedded submanifold is a subset of a larger manifold that is itself a manifold. This means that it has a well-defined notion of smoothness and can be described by coordinates in a neighborhood.

3. How are diffeomorphisms related to embedded submanifolds?

Diffeomorphisms are often used to transform or map one manifold onto another, and this can be used to show that two manifolds are diffeomorphic. Embedded submanifolds are often defined using diffeomorphisms, such as by embedding a smaller manifold into a larger one.

4. What are some applications of diffeomorphisms and embedded submanifolds?

Diffeomorphisms and embedded submanifolds are important concepts in differential geometry and have many applications in physics, such as in general relativity and fluid dynamics. They are also used in computer graphics and computer vision to model and analyze 3D objects.

5. Are there any limitations or challenges with using diffeomorphisms and embedded submanifolds?

One challenge with using diffeomorphisms and embedded submanifolds is that they can be difficult to visualize and manipulate, especially in higher dimensions. Additionally, finding a diffeomorphism between two manifolds can be a complex and computationally intensive task.

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