Let $(M,g)$ be a Riemannian manifold of dimension $n$ with Riemannian connection $\nabla,$ and let $p \in M.$ Show that there exists a neighborhood $U \subset M$ of $p$ and $n$ (smooth) vector fields $E_1,…,E_n \in \chi (U),$ orthonormal at each point of $U,$ s.t. at $p,$ $\nabla_{E_i}E_j(p)=0.$
What I've found so far;
Let $V$ be a normal neighborhood at $p.$ Let $(W,x(x_1,…,x_n))$ be a local coordinate system at $p$ s.t. $x(W) \subset V.$ Let $\{X_i:=\frac{\partial}{\partial x_i}\}$ be the standard basis of $T_pM.$ By Gram-Schmidt, there exists an orthonormal basis $\{E_i\}$ of $T_pM.$ Consider the orthonormal frame $E_i(t),$ by parallel transporting $\{E_i(0)=E_i\}$ along a a geodesic $\gamma :I \to M$ starting at $\gamma(0)=p$ and ending in $\gamma(1)=q,$ where $q \in V.$ This gives us an orthonormal frame at each point of $V.$ To satisfy the second condition, let $E_i(t)=\sum_{l}a_{li}X_l$ and $E_j(t)=\sum_{s}b_{sj}(t)X_s.$ Then
$$\nabla_{E_i}E_j=\sum_{k}(\sum_{l,s} a_{li}(t)b_{sj}(t)\Gamma^k_{ls}+E_i(b_{kj}))X_k,$$
where $\Gamma^k_{is}$ are Christoffel symbols.
To have $\nabla_{E_i}E_j(p)=0,$ we must have $\sum_{l,s}a_{li}(0)b_{sj}(0)\Gamma^k_{ls}+E_i(b_{kj})(p)=0$ for each $k.$
How can I conclude the argument?
Best Answer
Let $U=B_r(p)\subset M^n$ be a normal neighborhood. For each $q\in U$, there is a normalized geodesic $\gamma_q$ joining $p$ with $q$ (radial geodesic). Let $\{v_1,\ldots,v_n\}$ be an orthonormal basis of $T_pM$ and let $\{V_1,\ldots,V_n\}$ be their respective parallel transports along $\gamma_q$. For each $j=1,\ldots,n$, define the field $E_j$ by $$E_j(q)=V_j(d(p,q)),$$ where $d$ is the Riemannian distance. One has that $E_j$ is a $C^{\infty}$ field, because the curves $\gamma_q$ vary $C^{\infty}$ with $q$, in the sense that the EDO's of the geodesics $\gamma_q$ have their coefficients depending $C^{\infty}$ on $q$.
Now, consider $\sigma_i(s)$ the normalized geodesic such that $\sigma_i(0)=p$ and $$\sigma_i'(0)=v_i=V_i(0)=E_i(p).$$ One has, $$\nabla_{E_i}E_j(p)=\nabla_{E_i(p)}E_j=\nabla_{\sigma_i'(0)}E_j=\frac{D(E_j\circ\sigma_i)}{ds}(s)\Big|_{s=0}$$ Since $(E_j\circ\sigma_i)(s)=V_j(d(p,\sigma_i(s)))=V_j(s)$ is a parallel field along $\gamma_{\sigma_i(s)}=\sigma_i\big|_{[0,s]}$, we have that $$\nabla_{E_i}E_j(p)=\frac{D(E_j\circ\sigma_i)}{ds}(s)\Big|_{s=0}=\frac{DV_j}{ds}(0)=0.$$