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Concept
Diffeomorphism
Summary
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable.
Given two manifolds and , a differentiable map is called a diffeomorphism if it is a bijection and its inverse is differentiable as well. If these functions are times continuously differentiable, is called a -diffeomorphism.
Two manifolds and are diffeomorphic (usually denoted ) if there is a diffeomorphism from to . They are -diffeomorphic if there is an times continuously differentiable bijective map between them whose inverse is also times continuously differentiable.
Given a subset of a manifold and a subset of a manifold , a function is said to be smooth if for all in there is a neighborhood of and a smooth function such that the restrictions agree: (note that is an extension of ). The function is said to be a diffeomorphism if it is bijective, smooth and its inverse is smooth.
Hadamard-Caccioppoli Theorem
If , are connected open subsets of such that is simply connected, a differentiable map is a diffeomorphism if it is proper and if the differential is bijective (and hence a linear isomorphism) at each point in .
First remark
It is essential for to be simply connected for the function to be globally invertible (under the sole condition that its derivative be a bijective map at each point). For example, consider the "realification" of the complex square function
Then is surjective and it satisfies
Thus, though is bijective at each point, is not invertible because it fails to be injective (e.g. ).
Second remark
Since the differential at a point (for a differentiable function)
is a linear map, it has a well-defined inverse if and only if is a bijection. The matrix representation of is the matrix of first-order partial derivatives whose entry in the -th row and -th column is . This so-called Jacobian matrix is often used for explicit computations.
Third remark
Diffeomorphisms are necessarily between manifolds of the same dimension.
Official source
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Ontological neighbourhood