I've been trying to come up with a oordinate free formula of Christoffel symbols. For the Christoffel symbols of the first kind it's really easy. Since
[tex]\Gamma_{\lambda\mu\nu} = \frac{1}{2}\left( g_{\mu\lambda,\nu}+g_{\nu\lambda,\mu} - g_{\mu\nu,\lambda}\right) [/tex]
we can easily generalize the formula:
[tex]\Gamma\left(X,Y,Z\right) = \frac{1}{2}\left(Yg\left(X,Z\right)+Zg\left(X,Y\right)-Xg\left(Y,Z\right)\right)[/tex]
How can we generalize in this way the Christoffel symbols of the second kind [tex]{\Gamma^\lambda}_{\mu\nu} = g^{\lambda\sigma}\Gamma_{\lambda\mu\nu}[/tex]
I'm thinking that a way of doing it would be through some kind of contraction, but I'm not sure how since the Christoffel symbols of the first kind are not tensors to begin with.
[tex]\Gamma_{\lambda\mu\nu} = \frac{1}{2}\left( g_{\mu\lambda,\nu}+g_{\nu\lambda,\mu} - g_{\mu\nu,\lambda}\right) [/tex]
we can easily generalize the formula:
[tex]\Gamma\left(X,Y,Z\right) = \frac{1}{2}\left(Yg\left(X,Z\right)+Zg\left(X,Y\right)-Xg\left(Y,Z\right)\right)[/tex]
How can we generalize in this way the Christoffel symbols of the second kind [tex]{\Gamma^\lambda}_{\mu\nu} = g^{\lambda\sigma}\Gamma_{\lambda\mu\nu}[/tex]
I'm thinking that a way of doing it would be through some kind of contraction, but I'm not sure how since the Christoffel symbols of the first kind are not tensors to begin with.
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