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.
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 2 Given 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 3 Any 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
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 4 Let 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 5 The above element of is called the curvature of the connection , and is denoted .
Proposition 6 One 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 7 The 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
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,