The Riemann curvature tensor November 9, 2009Posted by Akhil Mathew in differential geometry, MaBloWriMo.
Tags: connections, curvature tensor
Today I will discuss the Riemann curvature tensor. This is the other main invariant of a connection, along with the torsion. It turns out that on Riemannian manifolds with their canonical connections, this has a nice geometric interpretation that shows that it generalizes the curvature of a surface in space, which was defined and studied by Gauss. When , a Riemannian manifold is flat, i.e. locally isometric to Euclidean space.
Rather amusingly, the notion of a tensor hadn’t been formulated when Riemann discovered the curvature tensor.
Given a connection on the manifold , define the curvature tensor by
There is some checking to be done to show that is linear over the ring of smooth functions on , but this is a straightforward computation, and since it has already been done in detail here, I will omit the proof.
The main result I want to show today is the following:
Proposition 1Let be a manifold with a connection whose curvature tensor vanishes. Then if is a surface with open and a vector field along , then
In other words, there is a kind of symmetry that arises in this case. This too can be proved by computing in a coordinate system.
More conceptually, here is a different argument. Assume first that is an immersion at some point , and extend locally to the vector field in a neighborhood of . Now
where are at least locally -related to . Similarly,
Since , their difference is
since is -related (at least in a neighborhood of ) to .
This was a short post to tie some things up. Tomorrow’s should be longer.