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<j<k}. 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'}.

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