2
$\begingroup$

In my lecture notes I have two definitions of the Christoffel symbols. The first is the smooth functions $\Gamma^k_{ij}: U\subseteq M\to\mathbb{R}$ defined for $i,j,k=1,2$ by

$\Gamma^k_{ij}=\dfrac{1}{2}\Sigma_{l=1,2}(g^{-1})^{lk}(\dfrac{\partial g_{jl}}{\partial u_i}+\dfrac{\partial g_{li}}{\partial u_j}-\dfrac{\partial g_{ij}}{\partial u_l})$

The second as $\nabla_{\frac{\partial}{\partial u_i}}(\dfrac{\partial}{\partial u_j})=\Sigma_k \Gamma^k_{ij}\dfrac{\partial}{\partial u_k}$ where $\nabla$ is the Levi-Civita connection.

I see that the first one is a good way of computing the Christoffel symbols, but I have no idea what it means and how it relates to the second definition. Could anybody try to explain how they are related and what exactly these symbols are for?

  • 0
    They're equivalent. If I were to try to prove that they were equivalent, I would write down the Kozsul formula in coordinates.2012-04-23
  • 1
    There are several equivalent ways to develop the theory, but one way to look at it is that your _first_ equation defines the Christoffel symbols, and your _second_ one then uses now-defined Christoffel symbols to define the Levi-Civita connection. (One then has to do some work to prove that this definition is invariant under coordinate change and satisfies the nice properties we want the connection to have).2012-04-23
  • 0
    @HenningMakholm: I actually think its much better to use the second as a defintion (since this works for connections in general), and then derive the first equation.2012-04-23

2 Answers 2