## The test case: flat Riemannian manifoldsNovember 12, 2009

Posted by Akhil Mathew in differential geometry, MaBloWriMo.
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Recall that two Riemannian manifolds ${M,N}$ are isometric if there exists a diffeomorphism ${f: M \rightarrow N}$ 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 ${\mathbb{R}^n}$.

Theorem 1 (The Test Case) The Riemannian manifold ${M}$ is locally isometric to ${\mathbb{R}^n}$ if and only if the curvature tensor vanishes. (more…)

## Identities for the curvature tensorNovember 11, 2009

Posted by Akhil Mathew in differential geometry, MaBloWriMo.
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It turns out that the curvature tensor associated to the connection from a Riemannian pseudo-metric ${g}$ has to satisfy certain conditions.  (As usual, we denote by $\nabla$ the Levi-Civita connection associated to $g$, and we assume the ground manifold is smooth.)

First of all, we have skew-symmetry $\displaystyle R(X,Y)Z = -R(Y,X)Z.$

This is immediate from the definition.

Next, we have another variant of skew-symmetry:

Proposition 1 $\displaystyle g( R(X,Y) Z, W) = -g( R(X,Y) W, Z)$  (more…)

## The Riemann curvature tensorNovember 9, 2009

Posted by Akhil Mathew in differential geometry, MaBloWriMo.
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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 ${R \equiv 0}$, 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 ${\nabla}$ on the manifold ${M}$, define the curvature tensor ${R}$ by $\displaystyle R(X,Y)Z := \nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z - \nabla_{[X,Y]} Z.$

There is some checking to be done to show that ${R(X,Y)Z}$ is linear over the ring of smooth functions on ${M}$, 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 ${M}$ be a manifold with a connection ${\nabla}$ whose curvature tensor vanishes. Then if ${s: U \rightarrow M}$ is a surface with ${U \subset \mathbb{R}^2}$ open and ${V}$ a vector field along ${s}$, then $\displaystyle \frac{D}{\partial x} \frac{D}{\partial y} V = \frac{D}{\partial y} \frac{D}{\partial x} V.$ (more…)