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Transparent connections over negatively curved surfaces

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  • 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 $M$.
    Mathematics Subject Classification: 53C25, 53C21, 58F17, 35J15.


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