The fundamental theorem of Riemannian geometry and the Levi-Civita connection November 10, 2009Posted by Akhil Mathew in differential geometry, MaBloWriMo.
Tags: connections, Levi-Civita connection, Riemannian metrics
Ok, now onto the Levi-Civita connection. Fix a manifold with the pseudo-metric . This means essentially a metric, except that 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 on compatible with .
The connection is called the Levi-Civita connection, even though Christoffel apparently discovered it first.
Recall that is compatible with if and only if for all . Since
by the fact that covariant differentiation commutes with contractions and satisfies the derviative identity, compatibility is equivalent to
for all .
Now assume and we have such a connection associated to . Then the connection is uniquely determined by a bunch of Christoffel symbols, which we will determine in terms of by a bit of elementary algebra. In other words, we just need to compute . Now
We can get two other equations by cyclic permutation:
So let , by symmetry. Let ; these are smooth real functions. These equations can be written
These are three linear equations in the unknowns . The system is nonsingular, so we get a unique solution, and consequently by nondegeneracy a unique possibility for the .
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 , choose to satisfy the system of three equations outlined above where . Then set , and we have a connection with since the vector fields are a frame (i.e. a basis at each tangent space on ). It is symmetric, since the torsion vanishes (by ) on pairs , and hence identically—since it is a tensor. Now to check for compatibility.
The difference of the two terms in (1) vanishes when are of the form . The vanishing holds generally because the difference of the two sides, which is , is a tensor. Hence compatibility follows.
So we have proved the theorem when is an open submanifold of (though not necessarily with the canonical metric ). In general, cover by open subsets diffeomorphic to an open set in . We get connections on compatible with .
I claim that . This is an easy corollary of uniquness. So we can patch the connections together to get the one Levi-Civita connection on . In more detail, the whole idea of patching works as follows. Given on and , choose a neighorhood containing but contained in some . Multiply by cutoff functions which are 1 close to but zero outside to get ; then set .