I will now review some differential geometry. Namely, I’ll recall what it means to have a connection in a complex vector bundle {E}, and construct its curvature as an {E}-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 GL_n). 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 {M}, we let {\mathcal{A}^1(M)} denote smooth {\mathbb{C}}-valued differential forms, and {\mathcal{A}(M)} denote smooth functions. Similarly, for a complex bundle {E \rightarrow M}, we let {E(M)} denote (smooth, {\mathbb{C}}-valued) sections.

2.1. Connections

Recall that a connection on a vector bundle {E} over a smooth manifold {M} is a {\mathbb{C}}-homorphism

\displaystyle \nabla: E(M) \rightarrow (\mathcal{A}^1 \otimes E)(M)

that maps global sections of {M} to global sections of {\mathcal{A}^1 \otimes E}, which satisfies the Leibnitz rule

\displaystyle \nabla (f s) = (df) s + f \nabla s, \quad s \in E(M), f \in \mathcal{A}(M).

This is essentially a way of differentiating sections of {E}, because for any vector field {X} on {M}, we can define {\nabla_X s}, the covariant derivative (with respect to this connection) of {s} in the direction of {X}. This satisfies:

  1. {\nabla_{fX}(s) = f \nabla_X s}.
  2. {\nabla_{X}(fs) = (Xf) s + f \nabla_X s}.

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 {E}, 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 {n}-dimensional vector bundle {E}, over an open subset {U \subset M}, is a family of sections {e_1, \dots, e_n \in E(U)} that forms a basis at each point; thus {\left\{e_1, \dots, e_n\right\}} defines an isomorphism of vector bundles between {E|_U} and the trivial {n}-dimensional bundle. Then, {\nabla} is determined over {U} by the elements {\nabla e_1, \dots, \nabla e_n \in (\mathcal{A}^1 \otimes E)(U)}. For, any section {s} of {E(U)} can be written as {s=\sum f_i e_i } for the {f_i} smooth functions, and consequently

\displaystyle \nabla s = \sum e_i (df_i) + \sum f_i \nabla e_i.

In other words, if we use the frame {\left\{e_i\right\}} to identify each section of {E(U)} with the tuple {\left\{f_i\right\}} such that {s = \sum f_i e_i}, then {\nabla} acts by applying {d} and by multiplying by a suitable matrix corresponding to the {\nabla e_i}. In view of this, we make:

Definition 2 Given a frame {\mathfrak{F} = \left\{e_1, \dots, e_n\right\}} over {U} and a connection {\nabla}, we define the {n}-by-{n} matrix {\theta(\mathfrak{F})} of 1-forms via

\displaystyle \nabla \mathfrak{F} = \theta(\mathfrak{F}) \mathfrak{F}.

In other words, {\nabla e_i = \sum_j \theta(\mathfrak{F})_{ij} e_j} for each {j}.

Note that {\theta} itself makes no reference to the bundle: it is simply a matrix of 1-forms. Given a frame {\mathfrak{F}}, and given {g: U \rightarrow \mathrm{GL}_n(\mathbb{C})}, we can define a new frame {g \mathfrak{F}} 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:

\displaystyle \nabla( g \mathfrak{F} )= (dg) \mathfrak{F} + g \nabla \mathfrak{F} = (dg) \mathfrak{F} + g \theta(\mathfrak{F}) \mathfrak{F}.

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

\displaystyle \theta(g\mathfrak{F}) = (dg) g^{-1} + g \theta(\mathfrak{F}) g^{-1}, \quad g : U \rightarrow \mathrm{GL}_n(\mathbb{C}). \ \ \ \ \ (1)

Conversely, if we have for each local frame {\mathfrak{F}} of a vector bundle {E \rightarrow M}, a matrix {\theta(\mathfrak{F})} of 1-forms as above, which satisfy the transformation law (1), then we get a connection on {E}.

Proposition 3 Any vector bundle {E \rightarrow M} admits a connection.

Proof: It is easy to see that a convex combination of connections is a connection. Namely, in each coordinate patch {U} over which {E} is trivial with a fixed frame, we choose the matrix {\theta} arbitrarily and get some connection {\nabla'_U} on {E|_U}. Let these various {U}‘s form an open cover {\mathfrak{A}}.

Then, we can find a partition of unity {\phi_U, U \in \mathfrak{A}} subordinate to {\mathfrak{A}}, and we can define our global connection via

\displaystyle \nabla = \sum_U \phi_U \nabla_U'.

\Box

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 {M} be a smooth manifold, {E \rightarrow M} a smooth complex vector bundle. Given a connection {\nabla} on {E}, the curvature is going to be a global section of {\mathcal{A}^2 \otimes \hom(E, E)}: in other words, the global differential 2-forms with coefficients in the vector bundle {\hom(E, E)}.

Proposition 4 Let {s } be a section of {E}, and {X, Y} vector fields. The map:

\displaystyle s, X, Y \mapsto R(X, Y, s) = ( \nabla_Y \nabla_X - \nabla_X \nabla_Y - \nabla_{[X, Y]})s

is a bundle map {E \rightarrow E}, and is {\mathcal{A}(M)}-linear in {X, Y}.

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

Since the quantity {R(X, Y, s)} is {\mathcal{A}(M)}-linear in all three quantities ({X, Y, s}), and clearly alternating in {X, Y}, we can think of it as a global section of the bundle {\mathcal{A}^2 \otimes \hom(E, E)}. Here, recall that {\mathcal{A}^2} is the bundle of 2-forms.

Definition 5 The above element of {(\mathcal{A}^2 \otimes \hom(E, E))(M)} is called the curvature of the connection {\nabla}, and is denoted {\Theta}.

We now wish to think of the curvature in another manner. To do this, we start by extending the connection {\nabla} to maps {\nabla: (E \otimes \mathcal{A}^p)(M) \rightarrow ( E \otimes \mathcal{A}^{p+1} )(M) }. The requirement is that the Leibnitz rule hold: that is,

\displaystyle \nabla(\omega s) = (d \omega) s + (-1)^p \omega \wedge \nabla s, \ \ \ \ \ (2)

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

Proposition 6 One can extend {\nabla} to maps {(E \otimes \mathcal{A}^p)(M) \rightarrow ( E \otimes \mathcal{A}^{p+1} )(M)} satisfying(2).

Given such an extension, we can consider the map

\displaystyle \nabla^2: E(M) \rightarrow (E \otimes \mathcal{A}^2(M)).

This is {\mathcal{A}(M)}linear. Indeed, we can check this by computation:

We want to now connect this {\mathcal{A}(M)}-linear map with the earlier curvature tensor.

Proposition 7 The vector-bundle map {\nabla^2} is equal to the curvature tensor {\Theta}.

Proof: We can work in local coordinates, and assume that {X,Y} are the standard commuting vector fields {\partial_i, \partial_j}. We want to show that, given a section {s}, we have

\displaystyle \nabla^2(s)(\partial_i, \partial_j) = \left(\nabla_{\partial_i} \nabla_{\partial_j} - \nabla_{\partial_j} \nabla_{\partial_i}\right) s\in E(M).

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

\displaystyle \nabla s = \sum_i dx_i \nabla_{\partial_i } s ,

and consequently

\displaystyle \nabla^2 s = \sum_{i,j} dx_j \wedge \nabla_{\partial_j} \left( dx_i \nabla_{\partial_i } s \right) .

This becomes, by the sign rules,

\displaystyle - \sum_{i,j} \nabla_{\partial_j} \nabla_{\partial_i} s dx_j \wedge dx_i = \sum_{i<j} ( \nabla_{\partial_j} \nabla_{\partial_i} - \nabla_{\partial_i} \nabla_{\partial_j})s dx_i \wedge dx_j.

It is easy to see that this, evaluated on {(\partial_i, \partial_j)}, gives the desired quantity. It follows that {\nabla^2} can be identified with {\Theta}. \Box

As a result, we may calculate the curvature in a frame. Let {\mathfrak{F} = \left\{e_1, \dots, e_n\right\}} be a frame, and let {\theta(\mathfrak{F})} be the connection matrix. Then we can obtain an {n}-by-{n} curvature matrix {\Theta(\mathfrak{F})} of 2-forms such that

\displaystyle \Theta(\mathfrak{F}) = \nabla^2 \mathfrak{F}.

The following result enables us to compute {\Theta(\mathfrak{F})}.

Proposition 8 (Cartan equation)

\displaystyle \Theta(\mathfrak{F}) = d \theta(\mathfrak{F}) - \theta(\mathfrak{F}) \wedge \theta(\mathfrak{F}). \ \ \ \ \ (3)

Note that {\theta(\mathfrak{F}) \wedge \theta(\mathfrak{F})} 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 {\nabla^2} acts on the frame {\left\{e_i\right\}}. Namely, with an abuse of notation:

\displaystyle \nabla^2 (\mathfrak{F}) = \nabla(\nabla \mathfrak{F}) = \nabla ( \theta(\mathfrak{F}) \mathfrak{F}) = d \theta(\mathfrak{F}) \mathfrak{F} - \theta(\mathfrak{F}) \wedge \left( \theta(\mathfrak{F}) \mathfrak{F} \right).

We have used the formula that describes how {\nabla} acts on a product with a form.

As a result, in the frame {\left\{e_i\right\}}, the matrix for {\nabla^2} is what was claimed. \Box

Finally, we shall need an expression for {d \Theta}. We shall state this in terms of a local frame.

Proposition 9 (Bianchi identity) With respect to a frame {\mathfrak{F}}, {d \Theta(\mathfrak{F}) = [\theta(\mathfrak{F}), \Theta(\mathfrak{F})]}.

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,

\displaystyle d \Theta(\mathfrak{F}) = d\left( d \theta(\mathfrak{F}) - \theta(\mathfrak{F}) \wedge \theta(\mathfrak{F}) \right) = -d \theta(\mathfrak{F}) \wedge \theta(\mathfrak{F}) + \theta(\mathfrak{F}) \wedge d \theta(\mathfrak{F}).

Similarly,

\displaystyle [\theta(\mathfrak{F}), \Theta(\mathfrak{F})] = [ \theta(\mathfrak{F}), d \theta(\mathfrak{F}) + \theta(\mathfrak{F}) \wedge \theta(\mathfrak{F})] = [\theta(\mathfrak{F}), d \theta(\mathfrak{F})]

because {[\theta(\mathfrak{F}), \theta(\mathfrak{F}) \wedge \theta(\mathfrak{F})] =0}. \Box

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