After we have connections on a vector bundle, we may consider its curvature just as what we did in Riemannian geometry. Furthermore, using the curvature, the Chern class is introduces to characterze vector bundles on a manifold. However, the Chern class turns out to be independent of the connection and hence gives us an important tool in the study of the topology of manifolds and bundles.
Curvature of a connection
Recall that a connection on a complex vector bundle is given as a sheaf homomorphism We see that is not a homomorphism of -modules, as is required to satisfy the Leibniz’ rule The connection can be extended into sheaf homomorphisms by where and are given on an open subset . This is well-defined as we can verify that We see that the generalized Leibniz’ rule is satisfied, that is for any and , we have
Thus we may consider the decompostion of with itself For a local section of and a local smooth function , we have This shows that is a homomorphism of -modules. Consequently, may be identified with a global section of the bundle , where is the bundle of endomorphisms of and is usually isdentified with . The -valued -form is called the curvature of the connection .
Suppose form a local frame of and with respect to this frame the connection matrix of is , i.e., We can express as a matrix of -forms locally by Direct computation yields that Thus i.e., The matrix is called the curvature matrix of with respect to the local frame . Consider another local frame with corresponding curvature matrix . If the relations between and is given by the matrix , i.e., then we can compute that This verifies the tensorial property of and .
Additional properties of the vector bundle and the connection yields addition properties of the curvature . If is a holomorphic vector bundle and is a complex connection, then considering a holomorphic frame of yields that , i.e., . If is further a hermitian vector bundle, then we may consider an orthonormal frame . With respect to this frame, we have and hence i.e., Thus implying that . Thus is of type and is skew-hermitian.
Given connections and curvature on some vector bundles, we can consider the induced connections and curvature on induced bundles, including direct sum, tensor product, dual bundle, and pullback.
Suppose and are vector bundles on with corresponding connections and curvature . Then we have a natural connection on the direct sum given as The corresponding curvature is clearly
Similarly we have a natural connection on the tensor product given as where and We can verify that the curvature of can be given as where we use similar identifications as above.
Now suppose is a vector bundle on with connection and curvature . The induced connection on the dual bundle is given by the formula where is a local section of and is a local section of . Thus With respect to a dual frame, we see that .
Further suppose is a smooth map. Then the pullback defines a connection on the pullback bundle . If is the connection matrix of with respect to the frame on an open subset , then the connection matrix of with respect to the frame on the open set is exactly . Thus we can see that the curvature of is nothing else than .
Chern classes of a vector bundle
In order to define the Chern classes of a vector bundle, we need to discuss a bit about functions of matrices which are invariant under conjugation.
Suppose is a homogeneous polynomial function in the entries of degree . We call an invariant polynomial if holds for any and .
The basic examples of such polynomials are the elementary symmetric polynomials of the eigenvalues of . Consider the polynomials given by the formula In particular, we have , , and is the sum of the determinants of -principal minors of . These are called elementary invariant polynomials.
In fact, any invariant polynomial can be expressed as a polynomial of these elementary invariant polynomials. Suppose is an invariant polynomial. Consider the polynomial which is symmetric in . Then there exists a polynomial such that where is the elementary symmetric polynomial in of degree . It follows that holds for any diagonalizable matrix , and hence holds for any matrix .
Now consider an -linear form We say that is invariant if holds for any and . We see that each invariant -linear form gives an invariant polynomial of degree by Conversely, it can be verified that each invariant polynomial of degree admits a symmetric invariant -form such that and are related as above. Such is actually unique and is called the polarization of .
Suppose is a complex vector bundle of rank and is a connection on with curvature . Consider an open subset with a local frame of on . With respect to this frame, the connection has a connection matrix and a curvatire matrix . Noting that the wedge product of forms of even degree is commutative, the expression gives a well-defined -form on for each invariant polynomial of degree . Since the curvature matrix with respect to different frames only differ by a conjugation, we see that with local expression is a well-defined -form on .
Theorem. Suppose is a complex vector bundle on a smooth manifold and is the curvature of a given connection on . If is an invariant polynomial of degree , then and the cohomology class in the de Rham cohomology group is independent of the choice of the connection on .
Before the proof of the theorem, we need some preparations concerning the extension of the connection and some further properties of the invariant -forms.
Recall that the connection induces a natural dual connection by the formula Then we have a connection as we have a natural connection on the tensor bundle . This actually extends to sheaf homomorphisms
Lemma (Bianchi identity). Suppose is a connection on the complex vector bundle with the curvature . Then
Proof: The action of the connection on the endomorphism bundle can be verified to be given by where is a local section of and is a local section of . It follow that for any local section ,
Suppose is a symmetric -linear invariant form, where is the rank of . Using the invariance of , we obtain a well-defined symmetric -linear invariant form This further induces a natural -linear map Moving to the section sheaf, we obtain a sheaf homomorphism
Lemma. Suppose is a connection on the complex vector bundle of rank and is a symmetric -linear invariant form on . Then for any -valued forms , we have
Proof: By the explicit formula of the extension of to , we can see it suffices to show the case , i.e., to show that where This can be treated locally, where we may assume the connection matrix of is given by . Then direct computation yields that where is an matrices of smooth functions. Since the Leibniz’ rule implys that it remains to show By the multi-linearilty, we only need to show that holds for a normal matrices .This formula is just a corollary of the invariance of if we consider the derivative of the constant function
Now we can prove the theorem.
Proof of the theorem: Suppose is the polarizaton of . Then we have by the preceding two lemmas, showing the first assertion.
To prove the second assertion, assume that is another connection on . For each , we can see that also defines a connection on . Suppose the curvature of the connection is . It suffices to show that is always a -exact -form. By direct computation in local frame we can verify that which impies that
The above theorem gives for each complex vector bundle a homomorphism from the graded algebra of invariant polynomials to the graded algebra given by where is the curvature of any connection on . This homomorphism is called the Chern-Weil homomorphism.
Let denote the elementary invariant polynomial of degree . We define the Chern form of the connection on by where is the curvature of . The Chern class is then defined by The total Chern class is the sum of the Chern classes where we set . If is a complex manifold, then we take the Chern classes of to be the Chern classes of its holomorphic tangent bundle .
By the independence of the Chern classes with the connection, we can see the Chern classes must be real. Indeed, consider an hermitian metric on and a connection on compatible with the hermitian structure. Then the corresponding curvature is skew-hermtitian, i.e., . Hence we have implying that .
Suppose and are complex vector bundles on , with curvature and . The curvature on the direct sum is given by and hence Thus
Suppose is a complex vector bundle of rank on with curvature and is a line bundle on with curvature . Then the curvature of is given by It follows that More generally, if is a complex vector bundle of rank , then
The curvature of the dual bundle is given by implying that It follows that
If is a smooth map, then the curvature on the pullback bundle is exactly . Thus the Chern classes of is given as