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The test case: flat Riemannian manifolds
*November 12, 2009*

*Posted by Akhil Mathew in differential geometry, MaBloWriMo.*

Tags: curvature tensor, flat manifolds, Riemannian metrics, test case

1 comment so far

Tags: curvature tensor, flat manifolds, Riemannian metrics, test case

1 comment so far

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…)

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Identities for the curvature tensor
*November 11, 2009*

*Posted by Akhil Mathew in differential geometry, MaBloWriMo.*

Tags: Bianchi identity, connections, curvature tensor, eponymy, Riemannian metrics

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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…)

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The Riemann curvature tensor
*November 9, 2009*

*Posted by Akhil Mathew in differential geometry, MaBloWriMo.*

Tags: connections, curvature tensor

3 comments

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 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…)