Uniqueness of smooth structure

In summary, the individual is seeking assistance in proving the Global Frobenius theorem and mentions the need to prove a lemma involving an involutive distribution and a collection of integral manifolds. They are looking for a place where this has been proven or help in proving it. No new findings or additional information has been provided.
  • #1
center o bass
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I'm looking to prove the Global Frobenius theorem, however in order to do so I need to prove the following lemma:

If ##D## is an involutive distribution and and ##\left\{N_\alpha\right\}## is collection of integral manifolds of ##D## with a point in common, then ##N = \cup_\alpha N_{\alpha}## has a unique smooth structure making it into connected integral manifold of ##D## in which each ##N_\alpha## is an open submanifold.

Do you know somewhere where it is proved? Or can you help me prove it?
 
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  • #2
I'm sorry you are not generating any responses at the moment. Is there any additional information you can share with us? Any new findings?
 

1. What is the significance of the uniqueness of smooth structure?

The uniqueness of smooth structure is a fundamental concept in mathematics, specifically in the field of differential geometry. It refers to the fact that there is only one way to define smoothness on a given set or manifold. This ensures consistency and allows for the development of powerful mathematical tools and techniques.

2. How is smoothness defined on a manifold?

Smoothness on a manifold is defined in terms of smooth functions, which are functions that have continuous derivatives of all orders. A manifold is said to have a smooth structure if it is homeomorphic to Euclidean space and has a smooth atlas, which is a collection of charts that cover the manifold and have smooth transition functions between them.

3. What is the difference between a smooth structure and a smooth manifold?

A smooth manifold is a mathematical space that is locally similar to Euclidean space, but a smooth structure refers to the specific way in which smoothness is defined on that manifold. A smooth manifold can have multiple smooth structures, but they will all be equivalent in terms of the smooth functions that can be defined on the manifold.

4. Can a non-smooth manifold have a smooth structure?

No, a non-smooth manifold cannot have a smooth structure. The definition of a smooth structure requires the manifold to be homeomorphic to Euclidean space, which is not possible for non-smooth manifolds. Additionally, the existence of a smooth structure is a defining characteristic of a manifold, so a non-smooth manifold would not meet the criteria.

5. How does the uniqueness of smooth structure impact other areas of mathematics?

The uniqueness of smooth structure has far-reaching implications in various areas of mathematics, including differential geometry, topology, and mathematical physics. It allows for the development of powerful tools and techniques for studying smooth manifolds and their properties, which have applications in fields such as robotics, computer graphics, and engineering.

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