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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Characteristic classes for the deformation of flat connections


Author: Huei Shyong Lue
Journal: Trans. Amer. Math. Soc. 217 (1976), 379-393
MSC: Primary 57D30
MathSciNet review: 0402774
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Abstract: In this paper, we study the secondary characteristic classes derived from flat connections. Let M be a differential manifold with flat connection $ {\omega _0}$. If f is a diffeomorphism of M, then $ {\omega _1} = {f^\ast}{\omega _0}$ is another flat connection. Denote by $ \alpha $ the difference of these two connections. Then $ \alpha $ and its exterior covariant derivative $ D\alpha $ are both tensorial forms on M. To each invariant polynomial $ \varphi $ of $ {\text{GL}}(n,{\text{R}})$, where $ n = \dim M,\varphi (\alpha ;D\alpha )$ is a globally defined form on M. The class $ \{ \varphi (\alpha ;D\alpha )\} \in H(M;{\text{R}})$ for $ \deg \varphi > 1$ gives rise to an obstruction of the deformability from $ {\omega _0}$ to $ {\omega _1}$. In particular, we prove that $ ( + )$ and $ ( - )$ connections, in the sense of E. Cartan, cannot be deformed to each other.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1976-0402774-3
PII: S 0002-9947(1976)0402774-3
Article copyright: © Copyright 1976 American Mathematical Society