Christoffel symbol

The Christoffel symbol describes the connection between nearby tangent spaces on a manifold and is used to define the covariant derivative. There is a unique Christoffel symbol associated with each metric, and so we can say that the Christoffel symbol measures the curvature of the manifold.

We write the Christoffel symbol as

\begin{align*} \Gamma^\lambda_{\mu\nu} &= \frac12 g^{\lambda\sigma} (\partial_\mu g_{\nu\sigma} + \partial_\nu g_{\sigma\mu} - \partial_\sigma g_{\mu\nu}) \\ \Gamma_{\lambda\mu\nu} &= \frac{1}{2} (\partial\mu g_{\nu\lambda} + \partial{\nu} g_{\lambda\mu} - \partial{\lambda} g_{\mu\nu}). \end{align*}

Or, equivalently

\begin{align*} \partial_\beta \vec e_\alpha = \Gamma^\mu_{\alpha\beta} \vec e_\mu. \end{align*}

In this form it is clear that the Christoffel connection encodes the variation in the basis vectors. This is a useful concept for definition of the covariant derivative.

Despite the index notation, the Christoffel symbols are not tensors. They do not transform like tensors, and, in fact, that is the point. The transformation of the Christoffel symbols introduces terms that perfectly cancel those introduced by the derivative, alowing us to define a covariant derivative.

The Christoffel symbols are symmetric in the lower indices in a coordinate basis. Several more useful identities related to the connection:

\begin{align*} \partial_\lambda g_{\mu\nu} &= \Gamma_{\mu\nu\lambda} + \Gamma_{\nu\mu\lambda} \\ g_{\mu\kappa} \partial_\lambda g^{\kappa\nu} &= -g^{\kappa\nu} \partial_\lambda g_{\mu\kappa} \\ \partial_\lambda g^{\mu\nu} &= -\Gamma^{\mu}_{\lambda\kappa} g^{\kappa\nu} - \Gamma^\nu_{\lambda\kappa} g^{\kappa\mu}. \end{align*}