Here is another way of obtaining the Christoffel symbols with the symetry imposed by the torsion free condition

$$ \Gamma^i_{k\ell}=\Gamma^i_{\ell k}. $$

This goes back to Riemann's Habillitation.

Suppose that $(M,g)$ is a Riemann manifold of dimension $N$, $p\in M$. By fixing an orthonormal frame of $T_pM$ we can find local coordinates $(x^1,\dotsc, x^N)$ near $p$ such that, $\newcommand{\pa}{\partial} $

$$ x^i(p)=0, \;\; g=\sum_{i,j} g_{ij}(x) dx^i dx^j, $$

$$g_{ij}(x)= \delta_{ij} +\sum_{i,j}\left(\sum_k\pa_{x^k}g_{ij}(0) x^k\right) dx^i dx^j + O(|x|^2). $$

In other words, in these coordinates,

$$ g_{ij}(x)=\delta_{ij} +O(|x|). $$

Riemann was asking whether one can find new coordinates near $p$ such that in these coordinates the metric $g$ satisfies $g_{ij}=\delta_{ij}$.

As a first step, we can ask whether we can find a new system of coordinates such that, in these coordinates the metric $g$ is described by

$$ g=\sum_{ij}\hat{g}_{ij} dy^idy^j, $$

where

$$\hat{g}(y)=\delta_{ij}+ O(|y|^2). \tag{1} $$

The new coordinates $(y^j)$ are described in terms of the old coordinates $(x^i)$ by a family of Taylor approximations

$$y^j= x^j + \frac{1}{2}\sum_{ij}\gamma^j_{\ell k} x^\ell x^k + O(|x|^3),\;\; \gamma^j_{\ell k}=\gamma^j_{k\ell}. $$

The constraint (1) implies

$$ \gamma^j_{\ell k}=\frac{1}{2}\left(\pa_{x^\ell}g_{jk}+\pa_{x^\ell}g_{jk}-\pa_{x^j}g_{\ell k}\right)_{x=0}. $$

We see that, in the $x$ coordinates

$$ \Gamma^i_{k\ell}(p)=\gamma^i_{k\ell}, $$

because $g^{ij}(p)=\delta^{ij}$.

It took people several decades after Riemann's work to realize that the coefficients $\Gamma^i_{k\ell}$ are related to parallel transport, and ultimately, to a concept of connection.

Ultimately, to my mind, the best explanation for the torsion-free requirement comes from Cartan's moving frame technique. The clincher is the following technical fact: given a connection $\nabla$ on $TM$ and a $1$-form $\alpha\in \Omega^1(M)$ then for any vector fields $X,Y$ on $M$ we have

$$d\alpha(X,Y)= X\alpha(Y)-Y\alpha(X)-\alpha([X,Y]) $$

$$= (\nabla_X\alpha)(Y)-(\nabla_Y\alpha)(X)+\alpha(\nabla_XY-\nabla_YX)-\alpha([X,Y]) $$

$$= (\nabla_X\alpha)(Y)-(\nabla_Y\alpha)(X)+\alpha\bigl(\;T_\nabla(X,Y)\;\bigr). $$

If the torsion is zero, the above equality looses a term, and one obtains rather easily Cartan's structural equations of a Riemann manifold.

scalar functions. $\endgroup$ – Willie Wong Feb 25 '13 at 9:51