In the 1957 paper, *On the differentiability of isometries*, Richard S. Palais gives a way to construct the tangent spaces of a Riemannian manifold using only its metric space structure (Theorem, p.1).

Specifically, given a Riemannian manifold $M$ and a point $p \in M$, the tangent space $T_p M$ equals the group of germs of pointed local similitudes $\mathbb{R} \to M$, $0 \mapsto p$. (Definitions below.)

Question:What are the metric space properties of Riemannian manifolds which allow this construction to work for Riemannian manifolds but not for arbitrary metric manifolds?

This construction *can't* work for arbitrary metric manifolds because any second-countable, Hausdorff, locally Euclidean space is metrizable, and yet there exist such spaces which are not homeomorphic to any smooth manifold, and in particular have no good definition of tangent spaces. If this construction worked generally, these manifolds would have tangent spaces; contradiction.

**Definitions:** Given metric spaces $(M_1,d_1)$ and $(M_2, d_2)$, a *similitude* is a bijective map $f:M_1 \to M_2$ such that, for a particular $r > 0$, for all $x,y \in M_1$, $d_2(f(x),f(y))= r \cdot d_1(x,y)$. An isometry is just the special case when $r = 1$.

A local similitude is the obvious analog of a local isometry (see Burago, Burago, Ivanov, *Metric Geometry*, Definition 3.4.1., p.78). Namely, a map $f: M_1 \to M_2$ is called a *(pointed) local similitude* at $x \in M_1$ (to $p \in M_2$) if $x \in M_1$ has an open neighborhood $U_x \subseteq M_1$ such that the (restriction of) $f$ is a similitude $U_x \to V_p$, where $V_p$ is some open neighborhood of $p$.

(Pointed here is solely meant to denote here that the point $p \in M_2$ under consideration is fixed in advance, as opposed to a "non-pointed" local similitude at $x \in M_1$, which could map onto *any* open set in $M_2$, not necessarily a neighborhood of the specified point $p \in M_2$. The terminology admittedly probably makes more sense when $f: M_1 \to M_1$ is a self-map, and $p = x$.)

Germ here is meant to denote the usual equivalence relation, where two functions belong to the same germ if and only if they agree on some open neighborhood of the point in question.

**Note:** Needless to say, Richard S. Palais uses different terminology. I am fairly confident, but not 100% confident, that my characterization is accurate/equivalent. (See here.) It is at the very least superficially similar to the definition of tangent spaces for smooth manifolds (germs of smooth maps $\mathbb{R} \to M$, $0 \mapsto p$ under an equivalence relation of smooth jets, see here). Tangent vectors are used to formalize notions of direction (see), and I had already been lead to germs of pointed local similitudes when trying to formalize the intuition of "direction" from Euclidean space (albeit ones $M \to M, p \mapsto p$ rather than $\mathbb{R} \to M, 0 \mapsto p$), see here (although it needs to be revised further).

*Even if* my attempted characterization of his result is not equivalent, I am still interested in the construction Richard S. Palais mentions for the tangent space of a Riemannian manifold, and would still like to know which metric space properties (e.g. strictly intrinsic metric? length space? existence of geodesics? etc.) allow the construction to work for Riemannian manifolds but not for arbitrary metric space manifolds. It would be interesting in particular to observe whether those metric space properties also hold for, e.g., Finsler manifolds.

**Note:** This website is for "research-level" mathematics. However, this question is with reference to a fairly old paper (1957), so I am not sure if it is on-topic. If not, please tell me. Note that there are at least two other questions (here and here) on MathOverflow which mention/discuss the paper.

This question seems at least tangentially related.

tangentially related<-- well done, sir/madam/etc. $\endgroup$