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.

For a smooth (for now, real) manifold , we let denote smooth -valued differential forms, and denote smooth functions. Similarly, for a complex bundle , we let denote (smooth, -valued) sections.

**2.1. Connections**

Recall that a **connection** on a vector bundle over a smooth manifold is a -homorphism

that maps global sections of to global sections of , which satisfies the **Leibnitz rule**

This is essentially a way of differentiating sections of , because for any vector field on , we can define , the **covariant derivative** (with respect to this connection) of in the direction of . This satisfies:

- .
- .

In fact, these two properties *characterize* a connection, as is easily checked. Moreover, one checks that the definition of a connection is *local*: that is, given a connection on , one gets connections on each restriction to an open neighborhood.

We can describe connections locally, in terms of *frames.* Recall that a **frame** of an -dimensional vector bundle , over an open subset , is a family of sections that forms a basis at each point; thus defines an isomorphism of vector bundles between and the trivial -dimensional bundle. Then, is *determined* over by the elements . For, any section of can be written as for the smooth functions, and consequently

In other words, if we use the frame to identify each section of with the tuple such that , then acts by applying and by multiplying by a suitable matrix corresponding to the . In view of this, we make:

Definition 2Given a frame over and a connection , we define the -by- matrix of 1-forms via

In other words, for each .

Note that itself makes no reference to the bundle: it is simply a matrix of 1-forms. Given a frame , and given , we can define a new frame by multiplying on the left. We would like to determine how a connection *transforms* with respect to a change of frame, so that we can think of a connection in a different way. Namely, we have:

where is considered as a matrix of 1-forms. As a result, we get the *transformation law*

Conversely, if we have for each local frame of a vector bundle , a matrix of 1-forms as above, which satisfy the transformation law (1), then we get a connection on .

Proposition 3Any vector bundle admits a connection.

*Proof:* It is easy to see that a convex combination of connections is a connection. Namely, in each coordinate patch over which is trivial with a fixed frame, we choose the matrix arbitrarily and get some connection on . Let these various ‘s form an open cover .

Then, we can find a partition of unity subordinate to , and we can define our global connection via

**2.2. Curvature**

We want to now describe the **curvature** of a connection. A connection is a means of differentiating sections; however, the differentiation may not satisfy the standard result for functions that mixed partials are equal. The curvature will be the measure of how much that fails. Let be a smooth manifold, a smooth complex vector bundle. Given a connection on , the curvature is going to be a global section of : in other words, the global differential 2-forms with coefficients in the vector bundle .

Proposition 4Let be a section of , and vector fields. The map:

is a bundle map , and is -linear in .

*Proof:* Calculation, typically done to define the Riemann curvature tensor in the case of the tangent bundle.

Since the quantity is -linear in all three quantities (), and clearly alternating in , we can think of it as a global section of the bundle . Here, recall that is the bundle of 2-forms.

Definition 5The above element of is called thecurvatureof the connection , and is denoted .

We now wish to think of the curvature in another manner. To do this, we start by extending the connection to maps . The requirement is that the Leibnitz rule hold: that is,

whenever is a -form and a global section. We can do this locally (by the above formula), and the local definitions necessarily glue. Thus:

Proposition 6One can extend to maps satisfying(2).

Given such an extension, we can consider the map

This is –*linear*. Indeed, we can check this by computation:

We want to now connect this -linear map with the earlier curvature tensor.

Proposition 7The vector-bundle map is equal to the curvature tensor .

*Proof:* We can work in local coordinates, and assume that are the standard commuting vector fields . We want to show that, given a section , we have

To do this, we should check exactly how the was defined. Namely, we have, by definition

and consequently

This becomes, by the sign rules,

It is easy to see that this, evaluated on , gives the desired quantity. It follows that can be identified with .

As a result, we may calculate the *curvature* in a frame. Let be a frame, and let be the connection matrix. Then we can obtain an -by- **curvature matrix** of 2-forms such that

The following result enables us to compute .

Proposition 8 (Cartan equation)

Note that is not zero in general! The reason is that one is working with *matrices* of 1-forms, not just plain 1-forms. The wedge product is thus the matrix product, in a sense.

*Proof:* Indeed, we need to determine how acts on the frame . Namely, with an abuse of notation:

We have used the formula that describes how acts on a product with a form.

As a result, in the frame , the matrix for is what was claimed.

Finally, we shall need an expression for . We shall state this in terms of a local frame.

Proposition 9 (Bianchi identity)With respect to a frame , .

Here the right side consists of matrices of differential forms, so we can talk about the *commutator.* We shall use this identity at a crucial point in showing that the Chern-Weil homomorphism below is even well-defined.

*Proof:* This is a simple computation. For, by Cartan’s equations,

Similarly,

because .

August 7, 2011 at 11:16 pm

Hum. It just so happens that I was thinking about how to define a noncommutative connection today. The definition in terms of the covariant derivative generalizes immediately, although from the noncommutative perspective the following seems to be more natural:

Given an algebra (the ring of functions on a noncommutative space) and a module (the space of sections of a bundle over this space) we can form the trivial square-zero extension (I don’t know what this is supposed to be, really). The trivial extension admits a natural grading in which has degree zero and has degree one. Then a graded derivation of this algebra descends to a derivation of , and a covariant derivative is precisely a section of this descent map! The curvature describes the extent to which such a section fails to be a morphism of Lie algebras.

August 8, 2011 at 7:50 am

Interesting. Yes, this sounds about right. You might also be interested by the theory of D-modules on varieties (which are the same as flat connections in the algebraic sense), though they’re usually not defined in the way you suggest.

August 8, 2011 at 11:02 am

[…] now with the preliminaries on connections and curvature established, and the Chern classes summarized, it’s time to see how they connect with one […]

May 2, 2012 at 11:59 pm

This is a brilliant article. Could you suggest any text that presents more on cartan’s formalism, especially its applications to riemannian geometry? Thanks.

May 14, 2012 at 8:36 pm

I’m not really sure. “Differential Analysis on Complex Manifolds” by Wells is pretty nice, as is Griffiths-Harris.