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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?

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1 Answer 1

up vote 2 down vote accepted

Let $(U,\phi)$ be a coordinate neighborhood of $p$ and let $g_{ij}$ and $\Gamma_{ij}^{k}$ denote the Riemannian metric tensors and the Christoffel symbols, respectively. If we recall that $g_{ij}=(E_i,E_j)$ and $\Gamma_{ij}^{k}=\nabla_{E_i}E_j(E_k)$ for all $1\leq i,j,k\leq n$ (where $M$ is a smooth $n$-manifold and $E_1,\dots,E_n$ are the coordinate frames on $(U,\phi)$), then we need only have $g_{ij}=\delta_{ij}$ on $U$ and $\Gamma_{ij}^{k}(p)=0$.

You are right in choosing a normal coordinate system at $p$, that is, choosing a normal coordinate neighborhood $(U,\phi)$ at $p$. Let me recall that this means choosing an orthonormal basis $F_1,\dots,F_n$ of the tangent space $T_p(M)$, choosing a star-shaped neighborhood of the origin in $T_p(M)$ which is diffeomorphically mapped onto $U$ under the exponential mapping $\text{exp}_p:D_p\to M$ (where $D_p$ is an open subset of $T_p(M)$ containing $0$) and defining $\phi=\exp_{p}^{-1}$ on $U$ (here we identify $T_p(M)$ with $\mathbb{R}^n$ by the linear isomorphism mapping $F_i$ onto $e_i$, $1\leq i\leq n$, where $e_1,\dots,e_n$ is the standard basis of $\mathbb{R}^n$).

In normal coordinates, the geodesics are of the form $y^i=a^it$ where $a_i$ is a constant for all $1\leq i\leq n$. If you prove this, then you should easily be able to see that $\Gamma_{ij}^{k}(p)=0$ for all $1\leq i,j,k\leq n$ by looking at the second-order ordinary differential equation of geodesics. You can also check that $g_{ij}(p)=\delta_{ij}$ for all $1\leq i,j\leq n$.

I hope this helps!

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Dear @ Amitesh Datta: you mean $g_{ij}=\delta_{ij},$ right? and how having $\Gamma^k_{ij}(p)=0$ would imply that $\nabla_{E_i}E_j(p)=0?$ –  Ehsan M. Kermani Feb 16 '12 at 8:21
    
Dear @ehsanmo, thank you for the correction (I have fixed it). The equalities $\Gamma_{ij}^{k}=\nabla_{E_i}E_j(E_k)$ are the definitions of the Christoffel symbols (for all $1\leq i,j,k\leq n$). For example, please see the relevant Wikipedia article: Christoffel symbols. –  Amitesh Datta Feb 16 '12 at 8:33
    
Oops! you're right. I was looking at my last expression and couldn't see it. –  Ehsan M. Kermani Feb 16 '12 at 8:51
1  
@ehsanmo No, you are right. However, the question is to find some coordinate neighborhood of $p$ in which the given equalities (concerning the Riemannian metric tensor and the Christoffel symbols) are valid; of course, these equalities will not hold in all coordinate neighborhoods (as you observed, however, they hold in a suitable normal coordinate neighborhood). (Moreover, although the Levi-Civita connection on a Riemannian manifold is uniquely determined by the Riemannian metric, its values on coordinate frames depend on the coordinate frames in question.) –  Amitesh Datta Feb 16 '12 at 11:28
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I have two issues with the construction and the answer: 1) Why does the parallel transport necessarily produce smooth(!) vector fields (the value at each point is the result a different differential equations...) 2) Using the Christoffel symbols implies that the frame {Ei} is the standard frame associated with the coordinates. This, if true, is not clear from the construction of the frame using parallel transport. Am I not seeing something? –  user38075 Aug 16 '12 at 21:58

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