Christoffel symbols in differential geometry

[effy21]

Homework Statement

I'm having trouble figuring out how to use Christoffel symbols. Apart from the first three terms here, I can't understand what's going on between line 3 and 4. What formulas/definitions are being used? How do you find the product of two chirstoffel symbols? Where are all the minuses coming from?

Homework Equations

The Attempt at a Solution

It looks like the product rule is needed when three terms are multiplied together. The indices p and l seem to be interchangeable.

 

[stevendaryl]

The reason for the minus sign is because there are two different rules for covariant derivatives, depending on whether the index is raised or lowered. I'm less comfortable with the comma and semi-colon notation, so let me use the notation I'm familiar with:

\nabla_i A^j = \partial_i A^j + \Gamma^j_{i k} A^k

where I'm using \nabla_i X to mean what you would write X_{; i} and \partial_i to mean what you would write X_{, i}

For tensors with lowered indices, you get a minus sign:

\nabla_i A_j = \partial_i A_j - \Gamma^k_{i j} A_k

For tensors with both kinds of indices, you get some minus signs and some plus signs:

\nabla_i A^{j}_{k} = \partial_i A^j_k + \Gamma^j_{i m} A^m_k - \Gamma^l_{i k} A^j_l

One other thing you have to remember is that in an expression such as \Gamma^j_{ik} A^k, the repeated index, k is a "dummy index", which means that you can just as well write it as \Gamma^j_{ip} A^p, or using any other symbol. What's important, though, is that you have to choose the "dummy" index so that it doesn't clash with any other index. So you can't choose i or j.