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 1
Letbe 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.
November 11, 2009 at 8:16 pm
[…] connections, curvature tensor, eponymy, Riemannian metrics trackback It turns out that the curvature tensor associated to the connection from a Riemannian pseudo-metric has to satisfy certain conditions. […]
November 12, 2009 at 10:45 pm
[…] isometric if there exists a diffeomorphism that preserves the metric on the tangent spaces. The curvature tensor (associated to the Levi-Civita connection) measures the deviation from flatness, where a manifold […]
August 21, 2011 at 9:07 am
[…] can associate with it a curvature form, which is an -valued 2-form; this is a generalization of the Riemann curvature tensor (as some computations that I don’t feel like posting here will show). In the case of a line […]