I heard this statement somewhere. Can anyone provide a reference (or explanation of why this is true)?
(I have also heard that the metric can be expanded as a power series in terms of the curvature tensor. Presumably these facts are closely related, but barring some kind of assumption about the metric being analytic I don't see how it follows immediately from the power series approximation...)
Thanks!
Edit: As it stands now, the statement I want to hear something about is something like: If there is a diffeo $f : M \to N$ pulling back the curvature tensors, then for each $a \in M$, there are isometric neighborhoods of $a$ and $f(a)$. Is this true / can someone recommend a reference?
I found a reference for a result of E. Cartan that I feel answers this questions in a satisfying way.
It seems to say that the curvature tensor is the obstruction to extending a linear isometry between tangent spaces to a local isometry on the manifold. It is a little complicated to state, because comparing curvature tensors on the two manifolds is done (not with pullbacks as I originally thought it might be), but by first parallel translating the 4 vectors back to the point on whose tangent space we have defined a linear isometry, applying that isometry, and then parallel translating along a certain geodesic on the image manifold until we are at the right point, and then using the curvature tensor there.
This is on page 156 in Do Carmo Riemannian Geometry.
$M$ and $N$ are Riemannian manifolds of diemnsion $n$, and let $p \in M$ and $q \in N$.
$i : T_p(M) \to T_q(N)$ is a choice of linear isometry, and $V \subset M$ is a normal neighbrohood of $p$ with the property that $exp_q$ is defined at $i \circ exp_p^{-1}(V)$. Define $f : V \to N$ by $f(q) = exp_{q} \circ i \circ exp_p^{-1}(q)$, for $q \in V$.
For each $q \in V$ there is a unique normalized geoedesic $\gamma [o,t] \to M$ with $\gamma(0) = p$ and $\gamma(t) = q$. (Since we chose $V$ to be a normal neighborhood of $p$). $P_t$ denotes the parallel transport along $\gamma$ from $\gamma(0)$ to $\gamma(t)$.
$\phi_t : T_s(M) \to T_{f(s)} N$ $\phi_t(v) = Q_t \circ i \circ P_t^{-1}$ for $v \in T_sM$, where $Q_t$ is parallel transport along the geodesic in $N$ with $\alpha(0) = q$ and $\alpha'(0) = i(\gamma'(0))$.
Then, if for all $s \in V$ and $x,y,u,v \in T_s(M)$:
$\langle R(x,y)u,v \rangle = \langle R(\phi_t(x), \phi_t(y)) \phi_t(u), \phi_t(v) \rangle$, then $f: V \to f(V)$ is a local isometry with $df_p = i$.