Christoffel symbol or Christoffel symbols?

Question

I didn't get if there's a difference between "Christoffel symbol" or "Christoffel symbols" being the second some sort of components of the first? I've always used the term Christoffel symbols, but a professor of my university made an unclear argument pointing out the difference. Is that actually true or something used to be true? Thanks in advance

Answer 1

I'm used to seeing it in the plural, Christoffel symbols. I've never really questioned why, but my thoughts are:

1. Not all of the indices on the entity make a tensor. For example, if you have Chirstoffel symbols of the second kind $\Gamma^a_{bc}$, this is a rank-2 tensor for each fixed value of $a$. The full $\Gamma^a_{bc}$ is not, however, a rank-3 tensor. In this interpretation, you get one symbol for each value of $a$.
2. Christoffel symbols are related to connection coefficients - usually the term "Christoffel symbols" is used for connection coefficients in a coordinate basis. The plural "symbols" then aligns with the plural "coefficients." You wouldn't say (singular) "coefficient" in that context.