Journal of Modern Dynamics (JMD)

Transparent connections over negatively curved surfaces

Pages: 311 - 333, Issue 2, April 2009      doi:10.3934/jmd.2009.3.311

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Gabriel P. Paternain - Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge CB3 0WB, United Kingdom (email)

Abstract: Let $(M,g)$ be a closed oriented negatively curved surface. A unitary connection on a Hermitian vector bundle over $M$ is said to be transparent if its parallel transport along the closed geodesics of $g$ is the identity. We study the space of such connections modulo gauge and we prove a classification result in terms of the solutions of a certain PDE that arises naturally in the problem. We also show a local uniqueness result for the trivial connection and that there is a transparent $SU(2)$-connection associated to each meromorphic function on $M$.

Keywords:  Complex Hermitian vector bundle, transparent connection, the Livsic theoremfor non-abelian cocycles, geodesic flow.
Mathematics Subject Classification:  53C25, 53C21, 58F17, 35J15.

Received: October 2008;      Available Online: May 2009.