Symplectic connections on a Riemann surface and holomorphic immersions in the Lagrangian homogeneous space

Biswas, Indranil ; Raina, A. K. (2008) Symplectic connections on a Riemann surface and holomorphic immersions in the Lagrangian homogeneous space Journal of Geometry and Physics, 58 (10). pp. 1417-1428. ISSN 0393-0440

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Official URL: http://linkinghub.elsevier.com/retrieve/pii/S03930...

Related URL: http://dx.doi.org/10.1016/j.geomphys.2008.06.001

Abstract

Let GrL⊂ Gr(n, V) be the space of all Lagrangian subspaces C2n of equipped with the standard symplectic form. Let X~ be a universal cover of a compact connected Riemann surface X. We consider all immersions f : X~→ GrL satisfying the following two conditions: (1) the map f is equivariant with respect to some homomorphism into Sp(2n,C) of the Galois group of the covering X~ → X, and (2) the symmetric bilinear form on the pullback, to X, of the tautological vector bundle over GrL is fiberwise nondegenerate. Two such maps are called equivalent if they differ by the action of some fixed element of Sp(2n,C). We prove that the equivalence classes of all such maps are bijectively parametrized by pairs of the form (P, (F,∇ ) where P is a projective structure on X and (F,∇ ) is an equivalence class of flat O(n,C)-connection on X. Two flat O(n,C)-bundles are equivalent if the corresponding flat PO(n,C)-bundles are isomorphic.

Item Type:Article
Source:Copyright of this article belongs to Elsevier Science.
Keywords:Symplectic Connection; Differential Operator; Lagrangian Subspace
ID Code:3646
Deposited On:18 Oct 2010 10:10
Last Modified:18 Oct 2010 10:10

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