Local isometry and Hausdorff dimension

Question

I am currently reading Falconer's book on Hausdorff dimension. My question is whether Hausdorff dimension is invariant under local isometry between smooth Riemannian manifolds? I think it should be yes, but I am not sure how to choose the covering( needed for calculation of dimension) using the local charts explicitly to get the invariance. Any help would be very much beneficial for my understanding.

Answer 1

Yes, it is invariant, and in fact this is true in much greater generality than manifolds.

If $X$ is a metric space, and $X$ is not separable, it is easy to verify that the Hausdorff dimension is infinite, and non-separability is of course preserved under homeomorphisms, so without loss of generality assume $X$ is separable (which we typically have in your context anyway).

Define the local Hausdorff dimension* $\newcommand{\dimloc}{\operatorname{dim}_{\text{loc}}}$ at a point $x\in X$ as

$$\dimloc(X,x)=\inf_{U\ni x}\dim(U),$$

where the infimum is taken over open neighborhoods $U$ of $x$.

Then it is easy to see from the Lindelöf property that $\dimloc(X,x)\leq d$ for all $x\in X$ if and only if $\dim(X)\leq d$, so that $\dim(X)=\sup_{x\in X}\dimloc(X,x)$. Since $\dimloc(X,x)$ is clearly invariant under local isometries, it follows $\dim(X)$ is as well.

Remark 1

We could replace “local isometry” with “locally bi-Lipschitz homeomorphism” and the invariance is still assured by the same argument. In particular, in the context of subsets of a (Riemannian) manifold, Hausdorff dimension is invariant under diffeomorphisms.

Remark 2

In general one needs a metric to define Hausdorff dimension, but from the preceding discussion we can see that a coherent definition for smooth, non-Riemannian manifolds can be given using the local dimension defined here, in local coordinates. Such a definition will automatically coincide with the usual Hausdorff dimension whenever said manifold is given any compatible Riemannian structure.

*Remark 3

"Local dimension" is a term I just made up for this answer, and is probably not the best name for this quantity, as it misleadingly might suggest that if $\dimloc(X,x)=d$, then there is a neighborhood of $x$ with dimension $d$, when in fact there need not be.