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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?

2 Answers 2

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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?$2012-02-16
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    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](http://en.wikipedia.org/wiki/Christoffel_symbols#Definition).2012-02-16
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    Oops! you're right. I was looking at my last expression and couldn't see it.2012-02-16
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    But Christoffel symbols are not invariant under the change of coordinate systems. Am I missing something?2012-02-16
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    @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.)2012-02-16
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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?2012-08-16
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    Just for future readers, and in response to user38075's comment: this answer shows that the Christoffel symbols corresponding to the normal coordinates vanish and that the corresponding coordinate fields are orthonormal at $p$. These coordinate fields are not however guaranteed to be orthonormal in a neighborhood of $p$. To get the orthonormal frame sought by the question one can apply Gram-Schmidt to the normal coordinate fields. You then have to prove though that the resulting frame satisfies $\nabla_{E_i}E_j(p) = 0$.2015-01-12
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    @ Amitesh Datta: Something in your argument about the vanishing of the Christoffel symbols bothers me. When you use the geodesic equation (in normal coordinates) you get: $v^T \Gamma ^k v =0$ for evey $k$. But this does not imply the matrix $\Gamma ^k = 0$ (For example a two-dimensional rotation by $90^0$ satisfy this).2015-06-09
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    I can't seem to be able to get $\Gamma_{ij}^k$ to vanish at $p$ without it vanishing everywhere else. You have $\sum_{ij} a^ia^j\Gamma_{ij}^k(q)=0$ for all $q$ and all $k$, and this should yield $\Gamma_{ij}^k(q)=0$, shouldn't it? What am I doing wrong?2016-03-18
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    @ZachBlumenstein all geodesics through $p$ are in the form of $y^i=a^i t$ but this doesn't hold for any other points in $U$, you should be careful dealing with base points2017-05-04
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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 a orthonormal basis of $T_pM$ and let $\{V_1,\ldots,V_n\}$ their respective parallel transports along of $\gamma_q$. 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 field $C^{\infty}$, 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 $C^{\infty}$ of $q$.

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 parallel field along of $\gamma_{\sigma_i(s)}=\sigma_i\big|_{[0,s]}$, we have that $$\nabla_{E_i}E_j(p)=\frac{D(E_j\circ\sigma_i)}{ds}(s)\Big|_{s=0}=\frac{DV_j}{ds}(0)=0.$$