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A diffeomorphism is a type of function between two different manifolds (or spaces) that is both smooth and invertible. In simpler terms, it is a function that maps one space onto another in a smooth and reversible way.
For a mapping to be considered a diffeomorphism, it must satisfy two main requirements: it must be a smooth function, meaning it has continuous derivatives of all orders, and it must be invertible, meaning there exists an inverse function that maps the second space back to the first.
To show that a mapping is a diffeomorphism, you must first prove that it is a smooth function by demonstrating that it has continuous derivatives of all orders. Then, you must show that it is invertible by finding the inverse function and proving that it also has continuous derivatives of all orders. You can also use the inverse function theorem, which states that a differentiable function with a non-zero Jacobian (determinant of the derivative) at a point is invertible at that point.
No, a diffeomorphism can only exist between spaces of the same dimension. This is because the inverse function must also have the same dimension as the original function, and if the two spaces have different dimensions, it is not possible for the inverse function to exist.
Yes, all diffeomorphisms are bijective, meaning they are both injective (one-to-one) and surjective (onto). This is because they are invertible, so every point in the second space has a corresponding point in the first space, and vice versa.