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?
Best Answer
A connection $\nabla$ on $TM$ can be defined abstractly as a map $\nabla: \mathfrak X(M) \times \mathfrak X(M) \to \mathfrak X(M), (X,Y) \mapsto \nabla_X Y$, where $\mathfrak X(M)$ is the set of vector fields on $X$, such that $\nabla$ is $C^\infty(M)$ linear in the $X$ variable, $\mathbb R$ linear in $Y$, and satisfies the Leibniz rule $$ \nabla_X fY = df(X) Y + f\nabla_X Y $$ for smooth functions $f$. If $\partial_{x_i}$ is a local coordinate chart for $TM$ then it is easy to see that $\nabla$ is completely determined by the vector fields $\nabla_{\partial_{x_i}} \partial_{x_j}$. Then $\Gamma_{ij}^k$ is by definition the coordinate functions of $\nabla_{\partial_{x_i}}\partial_{x_j}$. Note that these coefficients do not piece together to form a tensor since a connection is not tensorial (it is not $C^\infty$ linear in $Y$).
Now, by definition, the Levi-Civita connection is the unique connection on $TM$ that has no torsion and is compatible with the metric. From these properties it can be shown that the Christoffel symbols of this connection are given by the first equation you gave.