It is known that for a topological space with different metrics, the Hausdorff dimensions may not be equal in general.
For the case of manifolds, suppose $M$ is a $n$-manifold with a metric(distance), from https://math.stackexchange.com/questions/931628/hausdorf-dimension-of-a-manifold-of-dimension-n, we know that $dim_H M$ and $n$ may not be the same. But whether we can get a relation between $\dim_H M$ and $n$? I mean, whether we can prove that $\dim_H M\leq n$ or $\dim_H M\geq n$?
Unfortunately, I have no idea how we can get it directly from the definition of Hausdorff dimension.