December 22, 2012
Over the past couple of days I have been brushing up on introductory differential geometry. I’ve blogged about this subject a fair bit in the past, but I’ve never really had a good feel for it. I’d therefore like to make this post, and the next, a “big picture” one, rather than focusing on the technical details.
1. Curvature of a connection
Let be a manifold, and let be a vector bundle. Suppose given a connection on . This determines, and is equivalent to, the data of parallel transport along each (smooth) curve . In other words, for each such , one gets an isomorphism of vector spaces
with certain nice properties: for example, given a concatenation of two smooth curves, the parallel transport behaves transitively. Moreover, a homotopy of curves induces a homotopy of the parallel transport operators.
In particular, if we fix a point , we get a map
that sends a loop at to the induced automorphism of given by parallel transport along it. (Here we’ll want to take to consist of smooth loops; it is weakly homotopy equivalent to the usual loop space.) (more…)
August 7, 2011
I will now review some differential geometry. Namely, I’ll recall what it means to have a connection in a complex vector bundle , and construct its curvature as an -valued global 2-form.
Now there is a fancy, clean approach to the theory of connections and curvature on principal bundles over a group (and a vector bundle basically corresponds to one such over ). This approach is awesomely slick and highly polished: basically, it axiomatizes the intuitive idea that a connection is a way of identifying different fibers of a vector bundle (via parallel transport). So what is a connection on a principal bundle over a manifold? It’s a compatible system of defining whether tangent vectors are horizontal: the horizontal curves are those that correspond to a parallel transport. Then all the comparatively ugly index-filled results in the classical approaches get transformed into elegant, short results about Lie-algebra valued differential forms.
In fact, the whole Chern-Weil business can be developed using this formalism, and it becomes very slick. But I would like to do it in a slightly less fancy way, using the Cartan formalism: this essentially amounts to working in frames systematically. Here a frame is a local system of sections which is a basis for a vector bundle, and constitutes a generalized form of local coordinates. We can formulate the notions of connections and curvature in terms of frames (they’re systems of forms associated to each frame that transform in a certain way).
The theory has a super-optimal amount of index-pushing to it, but nonetheless, it is one I would like to gain comfort with, e.g. because Griffiths-Harris use it in their book. When one wants to actually prove concrete, specific results about certain types of manifolds (e.g. Kahler manifolds), it may be helpful to use local coordinates. An analogy: the theory of derived categories replaces the Grothendieck spectral sequence with the statement that the derived functor of the composite is the composite of the derived functors. But for concrete instances, the spectral sequence is still huge. (more…)
November 10, 2009
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…)
November 9, 2009
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:
Let be a manifold with a connection whose curvature tensor vanishes. Then if is a surface with open and a vector field along , then (more…)
November 8, 2009
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