(Geodesic frame). Let $M$ be a Riemannian manifold of dimension $n$ and let $p \in M$. Show that there exists a neighborhood $U \subset M$ of $p$ and $n$ vector fields $E_1,...,E_n \in X(U)$, orthonormal at each point of $U$, such that, at $p$, $\nabla_{E_i}E_j(p) = 0$. Such a family $E_i, i = 1,..., n,$ of vector fields is called a (local) geodesic frame at p.
Question: I don't know how I can to prove the step: $\nabla_{E_i}E_j(p)=0$.
I am using $\exp(te_i)$ as a geodesic for this.
Let $M$ be a Riemannian manifold of dimensin $n$ and $p\in M$.
Consider $U$ to be a convex neighbourhood around $p$. Choose an orthogonal basis $\{ X_1(p),\cdots , X_n(p)\}$ at $T_p(M)$. Define for each $q\in U$, $X_i(q)$ be the parralel transport of $X_i(p)$ with respect to the unique geodesic joining $p$ and $q$. As we know parralel transport is an isometry (which is essentially some exercise in Do Carmo) $\langle X_i,X_j\rangle= \delta_{ij}$. And $\nabla_{X_i}X_j(p)=0$ (because $d(exp_p)(0)$ is the identity map on $T_p(M)$, so follows from the construction) .
