
> 
> **Edit:** According to comment of Andre Henriques we revise the question:
>  In this question a fractal is a metric space whose topological and Hausdorff dimensions are different. So we would like that this possible counter example comes from a Riemannian structure not merely from a metric space view point.
> 
> 
> 


Is there an example of an analytic manifold $M$ with two real analytic Riemanian metrics $g\_1,g\_2$ such that $M$ has a compact subset which is a fractal set with respect to $g\_1$ but is not a fractal set with respect to $g\_2$?



Does fractallity depend on the Riemannian metric?
The answer is no. Any two Riemannian metrics when restricted to a compact set are bi-Lipschitz equivalent and bi-LIpschitz homeomorphism preserves the Hausdorff dimension.


