community,
I am concerned with measuring distances of systems under diffeomorphisms. Concrety, I consider a smooth diffeomorphism $\varphi: M\rightarrow N$ from the smooth differentiable manifold $M$ to $N$. The distance function $d$ is given in either $M$ or $N$, say $N$ for our purposes. I would like to infer a suitable distance function for the manifold $M$. My professor suggested the technique pull-back. However, that was not what I was looking for since, this would give me the distance the points have in $N$ with the points of $M$. I am looking for something that is able to express how the distribution of distances of all points in the manifold changed, whether cluster occur ect. However, how can I infer a distance function for that purpose?
I may just be looking for the right term or concept. Please, note that the dimensionalities of my manifolds usually exceed 10.
PS I am having a physics background but never got into relativity theory whatsoever.
Thank you so much for suggestions.
Cheers