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The test case: flat Riemannian manifolds November 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 tensor November 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 tensor November 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…)
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