The test case: flat Riemannian manifolds November 12, 2009Posted by Akhil Mathew in differential geometry, MaBloWriMo.
Tags: curvature tensor, flat manifolds, Riemannian metrics, test case
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Recall that two Riemannian manifolds are 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 is flat if it is locally isometric to a neighborhood of .
Theorem 1 (The Test Case) The Riemannian manifold is locally isometric to if and only if the curvature tensor vanishes. (more…)
Identities for the curvature tensor November 11, 2009Posted by Akhil Mathew in differential geometry, MaBloWriMo.
Tags: Bianchi identity, connections, curvature tensor, eponymy, Riemannian metrics
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It turns out that the curvature tensor associated to the connection from a Riemannian pseudo-metric has to satisfy certain conditions. (As usual, we denote by the Levi-Civita connection associated to , and we assume the ground manifold is smooth.)
First of all, we have skew-symmetry
This is immediate from the definition.
Next, we have another variant of skew-symmetry:
Proposition 1 (more…)
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 (more…)