Covariant derivative of the Christoffel symbol

[redstone]

Homework Statement

Is the covariant derivative of a Christoffel symbol equal to zero? It seems like it would be since it is composed of nothing but metrics, and the covariant derivative of any metric is zero, right?

 

[nicksauce]

This seems like a meaningless question. The covariant derivative acts on tensors, but the Christoffel symbols are not tensors.

 

[redstone]

That's the first thing that came to my mind, but then the covariant derivative can still act on all the terms of the Christoffel symbol, since it is composed of tensors, so it seemed like it might still be a meaningful question. With that in mind, would you still consider it to be meaningless?

 

[nicksauce]

You mean because it is built out of metrics? But it is built out of partial derivatives of metrics, which makes the terms no longer tensors.