The Chrystoffel Symbols

**fobos3** · February 14th 2009, 03:26 AM

Hi guys. Can someone explain to me what is the geometrical meaning of the Crystoffel symbols. I've tried wikipedia but to no avail.

P.S.
By Christoffel symbols I mean $\Gamma^\lambda_{\phantom{\lambda} \mu \nu} = \dfrac{1}{2}g^{\lambda \sigma}(g_{\sigma \mu ,\nu} + g_{\sigma \nu ,\mu} - g_{\mu \nu ,\sigma})$

**HallsofIvy** · February 14th 2009, 05:27 AM

Originally Posted by fobos3

Hi guys. Can someone explain to me what is the geometrical meaning of the Crystoffel symbols. I've tried wikipedia but to no avail.

P.S.
By Christoffel symbols I mean $\Gamma^\lambda_{\phantom{\lambda} \mu \nu} = \dfrac{1}{2}g^{\lambda \sigma}(g_{\sigma \mu ,\nu} + g_{\sigma \nu ,\mu} - g_{\mu \nu ,\sigma})$

The "covariant derivative" of a vector $A_k$ , $A_{m;n}$ is the ordinary derivative minus $\Gamma^k_{mn} A_k$ and so that last term essentially measures how much the change in a function is due to the coordinate system rather than the function. By the way, since the "ordinary derivative" is not a tensor, neither are the Christoffel symbols which is why some people prefer the notation $\left[\begin{array}{c}k \\ mn\end{array}\right]$ since it doesn't "look" so much like a tensor.

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