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…)
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…)
Symmetric connections, corrected version November 9, 2009Posted by Akhil Mathew in differential geometry, MaBloWriMo.
Tags: connections, corrections, symmetric connections, torsion tensor
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My post yesterday on the torsion tensor and symmetry had a serious error. For some reason I thought that connections can be pulled back. I am correcting the latter part of that post (where I used that erroneous claim) here. I decided not to repeat the (as far as I know) correct earlier part.
Proposition 1 Let be a surface in , and let be a symmetric connection on . Then (more…)
The torsion tensor and symmetric connections November 8, 2009Posted by Akhil Mathew in differential geometry, MaBloWriMo.
Tags: connections, symmetric connections, torsion tensor
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Today I will discuss the torsion tensor of a Koszul connection. It measures the deviation from being symmetric in a sense defined below.
Given a Koszul connection on the smooth manifold , define the torsion tensor by
The tubular neighborhood theorem November 5, 2009Posted by Akhil Mathew in differential geometry, MaBloWriMo.
Tags: collar neighborhood theorem, connections, exponential map, tubular neighborhood theorem
If is a manifold and a compact submanifold, then a tubular neighborhood of consists of an open set diffeomorphic to a neighborhood of the zero section in some vector bundle over , by which corresponds to the zero section.
Theorem 1 Hypotheses as above, has a tubular neighborhood. (more…)
Geodesics and the exponential map November 4, 2009Posted by Akhil Mathew in differential geometry, MaBloWriMo.
Tags: connections, exponential map, geodesics, ordinary differential equations
Ok, we know what connections and covariant derivatives are. Now we can use them to get a map from the tangent space at one point to the manifold which is a local isomorphism. This is interesting because it gives a way of saying, “start at point and go five units in the direction of the tangent vector ,” in a rigorous sense, and will be useful in proofs of things like the tubular neighborhood theorem—which I’ll get to shortly.
Anyway, first I need to talk about geodesics. A geodesic is a curve such that the vector field along created by the derivative is parallel. In local coordinates , here’s what this means. Let the Christoffel symbols be . Then using the local formula for covariant differentiation along a curve, we get
so being a geodesic is equivalent to the system of differential equations
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…)
Parallelism determines the connection November 2, 2009Posted by Akhil Mathew in differential geometry, MaBloWriMo.
Tags: connections, covariant derivatives, MaBloWriMo, parallelism
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I’m going to try participating in Charles Siegel’s MaBloWriMo project of writing a short post a day for a month. In particular, I’m categorizing yesterday’s post that way too. I’m making no promises about meeting that every day, but much of the material I talk about lends itself to bite-sized pieces anyway.
There is a nice way to tie together (dare I say connect?) the material yesterday on parallelism with the axiomatic scheme for a Koszul connection. In particular, it shows that connections can be recovered from parallelism.
So, let’s pick a nonzero tangent vector , where is a smooth manifold endowed with a connection , and a vector field . Then makes sense from the axiomatic definition. We want to make this look more like a normal derivative.
Now choose a curve with . Then I claim that
Covariant derivatives and parallelism November 1, 2009Posted by Akhil Mathew in differential geometry, MaBloWriMo.
Tags: connections, covariant derivatives, ordinary differential equations, parallelism
[Nobody should read this post without reading the excellent comments below. It turns out that thinking more generally (via connections on the pullback bundle) clarifies things. Many thanks to the (anonymous) reader who posted them. –AM, 5/16]
A couple of days back I covered the definition of a (Koszul) connection. Now I will describe how this gives a way to differentiate vector fields along a curve.
First of all, here is a minor remark I should have made before. Given a connection and a vector field , the operation is linear in over smooth functions—thus it is a tensor (of type (1,1)), and the value at a point can be defined if is replaced by a tangent vector at . In other words, we get a map , where denotes the space of vector fields. We’re going to need this below. (more…)