This may be a silly question as it seems one of the several Euclidean results in that carry over to Riemannian manifolds by passing in coordinates, but I suspect that the issue is subtler, and I am not an expert in geometric measure theory, so I'd rather ask.
The question is: does an analogue of Preiss' theorem on rectifiability hold for Riemannian manifolds? In other words, is the following statement true:
Let $M$ be a Riemannian manifold. Let $E \subset M$ be a Borel set with positive and finite $k$-dimensional Hausdorff measure, where $k >0$. Assume that the $k$-dimensional density $$ 0<\lim_{r\to 0}\frac{\mathcal{H}^k(E\cap B(x,r))}{r^k}<+\infty \qquad (\star) $$ (existence is one of the requirements) for $\mathcal{H}^k$-a.e. $x \in E$. Then $k$ is an integer, and $E$ is rectifiable, i.e. it can be covered up to a $\mathcal{H}^k$-null set by $C^1$ $k$-dimensional submanifolds.
Here, I stress, $\mathcal{H}^k$ is the $k$-dimensional Hausdorff measure associated with the Riemannian metric structure, and $B(x,r)$ is the Riemannian ball.
The Euclidean version of the statement is well-known, cf for example
De Lellis, Camillo, Rectifiable sets, densities and tangent measures, Zurich Lectures in Advanced Mathematics. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-044-9/pbk). vi, 126 p. (2008). ZBL1183.28006.
One then needs to show that if $(\star)$ holds then it implies its Euclidean counterpart for (almost) all points in charts. It is immediate to show that for any $x\in E$ where $(\star)$ holds, there exists a (normal) coordinate chart around $x$ such that the Euclidean counterpart of $(\star)$ holds at $x$, but what about other close points in the neighborhood?
Notice that, in a neighborhood of a given point $x$, and for any fixed coordinate chart $U$ around $x \in E$, the Riemannian distance is bi-Lipschitz equivalent to the Euclidean distance in these charts:
$$ c |y-z| \leq d(y,z) \leq C |y-z|, \qquad y,z \in U $$
for some $c,C>0$, depending on $U$. This only yields that
$$ 0<\liminf_{r\to 0} \frac{\mathcal{H}^k_{\mathbb{R}^n}(E\cap B_{\mathbb{R}^n}(x,r))}{r^k} \leq \limsup_{r\to 0} \frac{\mathcal{H}^k_{\mathbb{R}^n}(E\cap B_{\mathbb{R}^n}(x,r))}{r^k} < +\infty, $$
where $_{\mathbb{R}^n}$ means that we are considering the corresponding objects using the Euclidean coordinate structure. This of course does not prove the existence of the density in charts, and we cannot invoke Preiss' Euclidean result on $U$.
---- UPDATE AFTER COMMENTS ----
Of course one can choose coordinates such that the constants $c,C$ in the above bi-Lipschitz embedding in $\mathbb{R}^n$ are arbitrarily close to $1$. In this way one can show that the limsup and the liminf of the density in those local coordinates are very close, and then use the stronger version of Preiss' theorem, which says that there exists a constant $c(n,k)>1$ such that if $$ 0< \limsup_{r\to 0} \frac{\mathcal{H}^k_{\mathbb{R}^n}(E\cap B_{\mathbb{R}^n}(x,r))}{r^k} \leq c(k,n) \liminf_{r\to 0} \frac{\mathcal{H}^k_{\mathbb{R}^n}(E\cap B_{\mathbb{R}^n}(x,r))}{r^k} < \infty $$ for $\mathcal{H}^k_{\mathbb{R}^n}$-almost every $x$, then $E$ is $k$-rectifiable (even if the density in coordinates a priori might not exist). However, I would rather avoid using this rather complex quantitative result, and I do think that the density in local coordinates exists provided that $(\star)$ holds.
