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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Representations of Euler classes


Author: Howard Osborn
Journal: Proc. Amer. Math. Soc. 31 (1972), 340-346
MSC: Primary 57D20; Secondary 13C99
DOI: https://doi.org/10.1090/S0002-9939-1972-0300303-8
MathSciNet review: 0300303
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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}}$.


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DOI: https://doi.org/10.1090/S0002-9939-1972-0300303-8
Keywords: Pfaffians, Hodge operator, connections, curvature transformations, Euler classes, Avez-Chern theorem, Gauss-Bonnet theorem
Article copyright: © Copyright 1972 American Mathematical Society