Covariant derivative of the Christoffel symbol

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Homework Help Overview

The discussion revolves around the covariant derivative of the Christoffel symbol, exploring whether it is equal to zero. The context involves concepts from differential geometry and tensor calculus.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants are questioning the nature of the Christoffel symbols and their relationship with tensors, particularly regarding the application of the covariant derivative. There is a debate on whether the covariant derivative can be meaningfully applied to the Christoffel symbols, given their composition from metrics and partial derivatives.

Discussion Status

The discussion is active, with participants expressing differing views on the validity of the original question. Some participants suggest that the question may lack meaning, while others argue for its relevance based on the structure of the Christoffel symbols.

Contextual Notes

There is an ongoing examination of the definitions and properties of tensors and metrics, particularly in relation to the covariant derivative. Participants are considering the implications of the terms involved in the Christoffel symbols.

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

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