What information can be recovered from the induced metric of a Riemannian manifold?

Question

Note: In what follows, "metric" means "metric space metric", not "Riemannian metric".

Imagine you are playing a game with a friend. You choose a Riemannian manifold $M$, thus a point set $M$ with an associated topology, an associated metric, an associated tangent bundle, and an associated Riemannian metric on that tangent bundle.

Then you "delete" all of the information besides the metric (and the topology, since that is induced automatically by the metric), and give $M$ to your friend to analyze.

Question: In theory, how much of the information about the original Riemannian manifold $M$ could your friend recover from the metric space $M$?

Obviously your friend could use the topology to identify that the space is a manifold.

However, could your friend use the metric to recognize that the manifold is differentiable, and thus reconstruct the tangent bundle? And could your friend also use the metric to reconstruct the original Riemannian metric you had on the tangent bundle?

Note: A pointer to a reference will (more than) suffice for an answer -- I imagine that the answers to these questions are standard results in Riemannian geometry or at least metric geometry with long, technical proofs that are found in every textbook. But I am not familiar enough with either subject to be sure of this or to know where to find a reference answering these questions.

I know that in the case of $\mathbb{R}^n$, given a metric which was induced by an inner product, one can reconstruct the inner product from the metric. So I am curious how far analogous reasoning can extend to arbitrary Riemannian manifolds.

Answer 1

The answer to all of the above questions is affirmative, according to a 1957 paper by Richard S. Palais (link). The key construction in the argument used is the following:

Given the metric space $M$, for every point $x \in M$, define the set $M_x$ (called the set of geodesics at $x$ parametrized proportionally to arc length) such that $\sigma \in M_x$ if and only if:

1. $\sigma$ is a map of an interval $(-a, b)$ into $M$ where $a$ and $b$ are positive real numbers of $\infty$.
2. $\sigma(0) = x$
3. There is a real number $r$ such that, given any $t \in (a,b)$: $$d(\sigma(t), \sigma(t + t')) = r t' $$ for all sufficiently small $t'$.
4. $\sigma$ is not a proper restriction of a mapping satisfying the preceding properties.

(Note: I believe/imagine that condition 4. can be replaced by considering instead the set of germs of functions satisfying conditions 1-3. A rigorous proof is requested/required.)

Palais then proceeds to show that each $M_x$ has to be an $n$-dimensional real Hilbert space, thereby constructing a tangent bundle and an inner product at each point, and having shown that this agrees with the tangent bundle from the original Riemannian manifold, it follows that the inner product "varies sufficiently smoothly" between points.

Note that condition 3. implies that all maps $\sigma \in M_x$ are local similitudes $\mathbb{R} \to M$: given a $t_0$ sufficiently small, such that $|t'| \le t_0$ implies that 3. holds, then $\sigma(B_0(t_0)) \subseteq B_x(rt_0)$