What is curvature, really?

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.)

 

The Cartan formalism

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.

 

The fundamental theorem of Riemannian geometry and the Levi-Civita connection

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 .

 

The Riemann 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 1

Let be a manifold with a connection whose curvature tensor vanishes. Then if is a surface with open and a vector field along , then

 

Symmetric connections, corrected version

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 

 

The torsion tensor and symmetric connections

Today I will discuss the torsion tensor of a Koszul connection. It measures the deviation from being symmetric in a sense defined below.

Torsion

Given a Koszul connection on the smooth manifold , define the torsion tensor by

 

 

The 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.

 

Geodesics and the exponential map

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

Time to continue the story for covariant derivatives and parallelism, and do what I promised yesterday on 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.)

 

Parallelism determines the connection

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