Constructing a Smooth Diffeomorphism on a Manifold with Proper Open Set

In summary: M to itself. In summary, using a partition of unity and a complete vector field, we can construct a smooth non-trivial diffeomorphism from M to itself that has the identity map as its restriction on M-U.
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Homework Statement


[tex]M[/tex] is a smooth manifold, [tex]U \subset M[/tex] is a proper open set.
Show that there exists a smooth non-trivial diffeomorphism from [tex]M[/tex] onto itself which restriction on [tex]M - U[/tex] is identity ("identity outside [tex]U[/tex]").

Homework Equations



The Attempt at a Solution


If there exists a non-zero vector field that is zero outside [tex]U[/tex], in principle, its flow may be the required diffeomorphism. How do I construct the vector field? It is difficult because I need to work in local charts or is it trivial to see such vector field exists? a local flow is a local one-parameter group of local diffeomorphisms. When can it be extended to the whole [tex]M[/tex]? Should the vector field be complete?

There are too many questions, I don't know how to start, please point me a right direction.
 
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Firstly, it is important to understand the definitions involved in this problem. A smooth manifold is a mathematical structure that allows us to do calculus on it. It is a topological space that is locally Euclidean, meaning that it looks like Euclidean space in small neighborhoods. A proper open set is an open set that is not equal to the entire manifold. A diffeomorphism is a smooth bijective map between manifolds that has a smooth inverse. A non-trivial diffeomorphism is one that is not just the identity map.

Now, to solve this problem, we need to find a diffeomorphism from M to itself that satisfies the given conditions. One way to do this is by using a partition of unity. A partition of unity is a collection of smooth functions that sum up to 1 and are supported on a given open cover of a manifold. In this case, we can use a partition of unity to construct a smooth vector field on M that is zero outside of U. This vector field can then be used to define a flow, which is a one-parameter group of diffeomorphisms. This flow will be the required diffeomorphism, as it will be non-trivial and have the identity map as its restriction on M-U.

To construct the vector field, we can use a partition of unity that is subordinate to a cover of U. This means that each of the smooth functions in the partition is supported on a set in the cover, and the sum of the functions is equal to 1 on U. We can then use these functions to define a vector field on M that is zero outside of U. This vector field will be smooth, as it is constructed using smooth functions, and it will be non-zero, as it is equal to the sum of the functions in the partition. This vector field can then be used to define a flow, which will be a diffeomorphism from M to itself that satisfies the given conditions.

In order for the flow to be defined on the entire manifold, the vector field needs to be complete. A complete vector field is one that has a flow defined for all time. In this case, the vector field is complete because it is supported on a compact subset of M (U is proper), and the flow will be defined for all time. This means that the diffeomorphism will be defined for all time, and it will be the required smooth non-trivial diffeom
 

1. What is the diffeomorphism problem?

The diffeomorphism problem is a mathematical concept that deals with the existence and uniqueness of solutions to differential equations. It involves determining whether two differentiable manifolds are equivalent or "diffeomorphic" to each other.

2. Why is the diffeomorphism problem important?

The diffeomorphism problem has important implications in various fields of mathematics and physics, including differential geometry, topology, and general relativity. It helps us understand the behavior of solutions to differential equations and the structure of differentiable manifolds.

3. How is the diffeomorphism problem solved?

The diffeomorphism problem is typically solved by constructing a diffeomorphism, or a smooth and invertible map, between two manifolds. This can be done by finding a set of coordinates that transforms one manifold into the other, or by using more advanced mathematical techniques such as Lie groups and Lie algebras.

4. What are the challenges associated with the diffeomorphism problem?

One of the main challenges of the diffeomorphism problem is that it can be difficult to determine whether two manifolds are diffeomorphic, even when they look similar. In addition, constructing a diffeomorphism can be a complex and time-consuming task, especially for higher dimensional manifolds.

5. How does the diffeomorphism problem relate to other problems in mathematics?

The diffeomorphism problem is closely related to other problems in mathematics, such as the homeomorphism problem and the isomorphism problem. All of these problems involve determining whether two mathematical objects are equivalent or isomorphic to each other, and they often require similar techniques and approaches to solve.

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