I have a question regarding isometric spaces and isometries on these spaces in general.
Say I have two metric spaces, two sets X and Y each with their own intrinsic metric, which are isometric (1), ie. there is a distance preserving, bijective function $\phi: X\rightarrow Y$ (and vice versa in fact, in my case).
Let $I(X)$ and $I(Y)$ be the two sets of isometries $\phi_X :X \rightarrow X$ and $\phi_Y: Y\rightarrow Y$ respectively. With composition, these are obviously groups.
Now, in my book they state that, because of (1), the two groups $I(X)$ and $I(Y)$ are isomomorphic, but I have trouble understanding why this is. How would I even come up with a group homeomorphism between the two groups to begin with? And what are the consequences of such a fact? Does it mean that the isometries behave in the same way in either of the two spaces?
Best Answer
Take the map$$\begin{array}{rccc}\Psi\colon&I(X)&\longrightarrow&I(Y)\\&f&\mapsto&\phi\circ f\circ\phi^{-1},\end{array}$$whose inverse is$$\begin{array}{rccc}\Psi^{-1}\colon&I(Y)&\longrightarrow&I(X)\\&f&\mapsto&\phi^{-1}\circ f\circ\phi.\end{array}$$