We were deriving the geodesic equation in our differential geometry class, and it relied on the following identity:
$2 g_{cm,e} = g_{cm,e} + g_{ce,m}$
where $g_{ab}$ is the metric tensor, and commas indicate partial derivatives.
I've been trying to prove this identity, but I'm having difficulty in doing so. Can someone fill in the details, since the rest of the derivation (in defining the Christoffel symbol for example) depends on it!
Here are the class notes where this identity is used(towards the bottom of the page):
Thanks!
It has already been pointed out in the comments, that the formula you state is not true. In the actual text, however, they use $$ 2g_{cm,e}\dot{x}^m\dot{x}^e = (g_{cm,e} + g_{ce,m}) \dot{x}^m\dot{x}^e, $$ which is a much weaker statement. In the end, there's in fact nothing deep going on here; They are simply writing $$ 2g_{cm,e}\dot{x}^m\dot{x}^e = g_{cm,e}\dot{x}^m\dot{x}^e + g_{cm,e}\dot{x}^m\dot{x}^e $$ and renaming the labels $m\to e$, $e \to m$ in the second term (as explained in the text) to get $$ 2g_{cm,e}\dot{x}^m\dot{x}^e = g_{cm,e}\dot{x}^m\dot{x}^e + g_{ce,m}\dot{x}^e\dot{x}^m = g_{cm,e}\dot{x}^m\dot{x}^e + g_{ce,m}\dot{x}^m\dot{x}^e = (g_{cm,e} + g_{ce,m}) \dot{x}^m\dot{x}^e $$