Ok, now onto the Levi-Civita connection. Fix a manifold ${M}$ with the pseudo-metric ${g}$. This means essentially a metric, except that ${g}$ as a bilinear form on the tangent spaces is still symmetric and nondegenerate but not necessarily positive definite. It is still possible to say that a pseudo-metric is compatible with a given connection.

This is the fundamental theorem of Riemannian geometry:

Theorem 1 There is a unique symmetric connection ${\nabla}$ on ${M}$ compatible with ${g}$.

The connection is called the Levi-Civita connection, even though Christoffel apparently discovered it first.

Recall that ${g}$ is compatible with ${\nabla}$ if and only if ${\nabla_X g = 0}$ for all ${X}$. Since $\displaystyle X(g(X_1,X_2)) = (\nabla_X g)(X_1,X_2) + g(\nabla_X X_1, X_2) + g(X_1, \nabla_X X_2) ,$

by the fact that covariant differentiation commutes with contractions and satisfies the derviative identity, compatibility is equivalent to $\displaystyle X (g (X_1,X_2)) = g(\nabla_X X_1, X_2) + g(X_1, \nabla_X X_2),\ \ \ \ \ (1)$

for all ${X,X_1, X_2}$.

Now assume ${M \subset \mathbb{R}^n}$ and we have such a connection associated to ${g}$. Then the connection is uniquely determined by a bunch of Christoffel symbols, which we will determine in terms of ${g}$ by a bit of elementary algebra. In other words, we just need to compute ${\nabla_{\partial_i} \partial_j}$. Now $\displaystyle \partial_k g( \partial_i, \partial_j) = g( \nabla_{\partial_k} \partial_i, \partial_j) + g( \partial_i, \nabla_{\partial_k}\partial_j).$

We can get two other equations by cyclic permutation: $\displaystyle \partial_i g( \partial_j, \partial_k) = g( \nabla_{\partial_i} \partial_j, \partial_k) + g( \partial_j, \nabla_{\partial_i}\partial_k)$ $\displaystyle \partial_j g( \partial_k, \partial_i) = g( \nabla_{\partial_j} \partial_k, \partial_i) + g( \partial_k, \nabla_{\partial_j}\partial_i)$

So let ${S_{ij} := \nabla_{\partial_i} \partial_j = \nabla_{\partial_j} \partial_i}$, by symmetry. Let ${T_{ijk} := \partial_i g( \partial_j, \partial_k)}$; these are smooth real functions. These equations can be written $\displaystyle T_{kij} = g( S_{ik}, \partial_j) + g( S_{jk}, \partial_i)$ $\displaystyle T_{ijk} = g( S_{ij}, \partial_k) + g( S_{ik}, \partial_j)$ $\displaystyle T_{jki} = g( S_{jk}, \partial_i) + g( S_{ij}, \partial_k)$

These are three linear equations in the unknowns ${g( S_{ik}, \partial_j), g( S_{jk}, \partial_i), g( S_{ij}, \partial_k)}$. The system is nonsingular, so we get a unique solution, and consequently by nondegeneracy a unique possibility for the ${S_{ij}}$.

We have just shown the uniqueness assertion of the theorem, which is local. Connections restrict to connections on open subsets.

Now for existence. Still with the hypothesis on ${M}$, choose ${S_{ij}}$ to satisfy the system of three equations outlined above where ${i. Then set ${S_{ji} := S_{ij}}$, and we have a connection ${\nabla}$ with ${\nabla_{\partial_i} \partial_j := S_{ij}}$ since the vector fields ${\partial_i}$ are a frame (i.e. a basis at each tangent space on ${M}$). It is symmetric, since the torsion ${T}$ vanishes (by ${S_{ij}=S_{ji}}$) on pairs ${(\partial_i,\partial_j)}$, and hence identically—since it is a tensor. Now to check for compatibility.

The difference of the two terms in (1) vanishes when ${X,X_1,X_2}$ are of the form ${\partial_i}$. The vanishing holds generally because the difference of the two sides, which is ${(\nabla_X g)(X_1,X_2)}$, is a tensor. Hence compatibility follows.

So we have proved the theorem when ${M}$ is an open submanifold of ${\mathbb{R}^n}$ (though not necessarily with the canonical metric ${\sum_{i=1}^n dx_i \otimes dx_i}$). In general, cover ${M}$ by open subsets ${U_i}$ diffeomorphic to an open set in ${\mathbb{R}^n}$. We get connections ${\nabla_i}$ on ${U_i}$ compatible with ${g|_{U_i}}$.

I claim that ${\nabla_i|_{U_i \cap U_j} = \nabla_j|_{U_i \cap U_j}}$. This is an easy corollary of uniquness. So we can patch the connections together to get the one Levi-Civita connection on ${M}$. In more detail, the whole idea of patching works as follows. Given ${X,Y}$ on ${M}$ and ${p \in M}$, choose a neighorhood ${U}$ containing ${p}$ but contained in some ${U_i}$. Multiply ${X,Y}$ by cutoff functions which are 1 close to ${p}$ but zero outside ${U}$ to get ${X',Y'}$; then set ${\nabla_X Y := \nabla_{i, X'}Y'}$.