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To Show A Function is A Diffeomorphism

caffeinemachine

Well-known member
MHB Math Scholar
Mar 10, 2012
834
A rigid motion in $\mathbb R^n$ is a function $:\mathbb R^n\to \mathbb R^n$ such that $||L(x)-L(y)||=||x-y||$ for all $x,y\in \mathbb R^n$.

Let $G$ be the set of all the rigid motions in $\mathbb R^n$ and $p_1,\ldots,p_m\in R^n$.

Define a map $\phi:G\to (\mathbb R^n)^m$ as $\phi(L)=(Lp_1,\ldots,Lp_m)$.

From rigid transformation - To Prove That a Certain Set is a Manifold - Mathematics Stack Exchange I came to know that $\phi$ is a diffeomorphism.

Now I am not sure how to show that $\phi$ is a diffeomorphism. May be first we need to see $G$ as a subset of some Euclidean space, which doesn't seem to be too hard since any member of $G$ can be represented by a matrix.

But I am not sure how to proceed from here.

Can anybody help?