Chern-, Pontrjagin-, and Euler- characteristic forms
We spell out the formulas for the images under the Chern-Weil homomorphism of the Chern classes, Pontrjagin classes and Euler classes as characteristic forms over smooth manifolds.
Preliminaries
Let be a smooth manifold.
Write
for the commutative algebra over the real numbers of even-degree differential forms on , under the wedge product of differential forms. This is naturally a graded commutative algebra, graded by form degree, but since we consider only forms in even degree it is actually a plain commutative algebra, too, after forgetting the grading.
Let be a semisimple Lie algebra (such as or ) with Lie algebra representation over the complex numbers of finite dimension (for instance the adjoint representation or the fundamental representation), hence a homomorphism of Lie algebras
to the linear endomorphism ring , regarded here through its commutator as the endomorphism Lie algebra of .
When regarded as an associative ring this is isomorphic to the matrix algebra of square matrices
(2)
The tensor product of the -algebras (1) and (2)
is equivalently the matrix algebra with coefficients in the complexification of even-degree differential forms:
The multiplicative unit
in this algebra is the smooth function (differential 0-forms) which is constant on the identity matrix and independent of .
Given a connection on a -principal bundle, we regard its -valued curvature form as an element of this algebra
The total Chern form is the determinant of the sum of the unit (3) with the curvature form (4), and its component in degree , for , is the th Chern form :
By the relation between determinant and trace, this is equal to the exponential of the trace of the logarithm of , this being the exponential series in the trace of the Mercator series in :
Setting in these expressions (5) yields the total Pontrjagin form with degree=-components the Pontrjagin forms :
Hence the first couple of Pontrjagin forms are
(See also, e.g., Nakahara 2003, Exp. 11.5)
For and with the curvature form again regarded as a 2-form valued -square matrix
the Euler form is its Pfaffian of this matrix, hence the following sum over permutations with summands signed by the the signature :
The first of these is, using the Einstein summation convention and the Levi-Civita symbol:
(See also, e.g., Nakahara 2003, Exp. 11.7)