Representations of Euler classes

Abstract:

For any endomorphism $K$ of an oriented module $F$ with inner product there is an element ${\text {pf }}K$ in the ground ring $R$, a constant multiple of the classical pfaffian in the case $F = {R^{2n}}$. If $R$ is the algebra of even-dimensional differential forms on a smooth manifold, and if $F$ is the tensor product of $R$ and the module of sections of an oriented $2n$-plane bundle, then any connection in the bundle induces a curvature transformation $K:F \to F$ for which ${(4\pi )^{ - n}}{\text {pf }}K$ represents the Euler class. Properties of Euler classes are immediate consequences of corresponding properties of ${\text {pf}}$.