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The Riemann curvature tensor

Posted by Akhil Mathew under

differential geometry,

MaBloWriMo | Tags:

connections,

curvature tensor |

[3] Comments
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**

*Let 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.

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