The connection [tex]\nabla[/tex] is defined in terms of its action on tensor fields. For example, acting on a vector field Y with respect to another vector field X we get(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\nabla_X Y = X^\mu ({Y^\alpha}_{,\mu} + Y^\nu {\Gamma^\alpha}_{\mu\nu})e_\alpha

= X^\mu {Y^\alpha}_{;\mu}e_\alpha[/tex]

and we call [tex]{Y^\alpha}_{;\mu}={Y^\alpha}_{,\mu} + Y^\nu {\Gamma^\alpha}_{\mu\nu}[/tex] the covariant derivative of the components of Y. We can similarly form the covariant derivative of the components of any rank tensor, by including other appropriate terms with the connection coefficients.

So what does it mean to take the covariant derivative of the connection coefficients themselves? They are not components of a tensor? I have just come across a reference to [tex]{\Gamma^\alpha}_{\mu\nu;\lambda}[/tex] and don't know what to do with it.

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# Covariant derivative of connection coefficients?

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