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…)
Tags: connections, Levi-Civita connection, Riemannian metrics
Ok, now onto the Levi-Civita connection. Fix a manifold with the pseudo-metric . This means essentially a metric, except that as a bilinear form on the tangent spaces is still symmetric and nondegenerate but not necessarily positive definite. It is still possible to say that a pseudo-metric is compatible with a given connection.
This is the fundamental theorem of Riemannian geometry:
Theorem 1 There is a unique symmetric connection on compatible with . (more…)
Covariant derivatives and parallelism for tensors November 3, 2009Posted by Akhil Mathew in differential geometry, MaBloWriMo.
Tags: connections, covariant derivatives, parallelism, Riemannian metrics, tensors
Fix a smooth manifold with a connection . Then parallel translation along a curve beginning at and ending at leads to an isomorphism , which depends smoothly on . For any , we get isomorphisms depending smoothly on . (Of course, given an isomorphism of vector spaces, there is an isomorphism sending —the important thing is the inverse.) (more…)
Riemannian metrics and connections October 27, 2009Posted by Akhil Mathew in differential geometry.
Tags: Christoffel symbols, connections, Riemannian metrics
Wow. Blogging is definitely way harder during the academic year.
Ok, so I’m aiming to change things around a bit here and take a break from algebraic number theory to do some differential geometry. I’ll assume basic familiarity with what manifolds are, the tangent bundle and its variants, but generally no more. I eventually want to get to some real theorems, but this post will focus primarily on definitions.
A Riemannian metric on a smooth manifold is defined as a covariant symmetric 2-tensor such that for all . For convenience, I will write for . In other words, a Riemannian metric is a collection of (positive) inner products on each of the tangent spaces such that if are (smooth) vector fields, the function defined by taking the inner product at each point, is smooth. There are several ways to get Riemannian metrics: (more…)