April  2009, 3(2): 311-333. doi: 10.3934/jmd.2009.3.311

Transparent connections over negatively curved surfaces

1. 

Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge CB3 0WB, United Kingdom

Received  October 2008 Published  May 2009

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 $M$.
Citation: Gabriel P. Paternain. Transparent connections over negatively curved surfaces. Journal of Modern Dynamics, 2009, 3 (2) : 311-333. doi: 10.3934/jmd.2009.3.311
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