## Geodesic Normal Coordinates, Gauss lemma and Identity

Suppose ${p}$ is a point on ${n-}$dim manifold ${M}$. Consider the exponential map near ${p}$, at suitable neighborhood any point within it can be expressed uniquely as

$\displaystyle \text{exp}(x^1e_1+x^2e_2+\cdots+x^ne_n)$

where ${\{e_i\}}$ is an orthonormal basis. Then ${\{x^i\}}$ can be coordinate around ${p}$. Let us deduce useful identities using this coordinate. Obviously, the geodesic in this coordinate will be

$\displaystyle \gamma(t)=tv, \quad v\in T_pM$

In particular ${v=e_i}$,

$\displaystyle g_{ij}(p)=\langle \frac{\partial }{\partial x^i}(0),\frac{\partial }{\partial x^j}(0)\rangle=\langle e_i,e_j\rangle=\delta _{ij}$

The geodesic equation will be

$\displaystyle \Gamma^k_{ij}(\gamma(t))v^iv^j=0$

Multiplying ${t^2}$, we get

$\displaystyle \Gamma^k_{ij}(\gamma(t))x^ix^j=0$

Since the tangent vector has constant length along ${\gamma(t)}$, i.e.

$\displaystyle g_{ij}(\gamma(t))v^iv^j=g_{ij}(p)v^iv^j=v^iv^i$

Multiplying ${t^2}$, it is equivalent to say

$\displaystyle g_{ij}x^ix^j=x^ix^i$

Furthermore, recalling the definition of Christoffel symbol, one can get

$\displaystyle \frac{1}{2}(\partial_j g_{ik}+\partial_i g_{jk}-\partial_k g_{ij})x^ix^j=0$

Or equivalently

$\displaystyle \partial_j g_{ik}x^ix^j=\frac{1}{2}\partial_kg_{ij}x^ix^j=\frac{1}{2}\partial_k(g_{ij}x^ix^j)-g_{kj}x^j=x^k-g_{kj}x^j$

While on the left hand side

$\displaystyle \partial_j g_{ik}x^ix^j=\partial_j(g_{ik}x^i)x^j-g_{ik}x^i$

Combining this two facts,

$\displaystyle \partial_j(g_{ik}x^i)x^j=x^k$

$\displaystyle \partial_j(g_{ik}x^i-x^k)x^j=0$

This means along ${\gamma(t)}$,

$\displaystyle \frac{d}{dt}(g_{ik}x^i-x^k)=0$

Since at ${p}$, ${g_{ik}x^i-x^k=0}$, one get

$\displaystyle g_{ik}x^i=x^k$

on any point in the neighborhood. This identity is called Gauss Lemma.

Suppose ${g(x)=\text{exp}(h(x))}$, ${h_{ij}(x)}$ is a symmetric 2-tensor. Then the above identity means ${h_{ij}x^j=0}$.