Mm Obius Structures and Two Dimensional Einstein-weyl Geometry

Abstract

Conformal 2-manifolds possess a fascinatingly rich and elegant theory which can be viewed in many ways: it is the theory of Riemann surfaces in complex analysis, or of complex curves in algebraic geometry. In this paper, a purely diierential geometric point of view will be taken, the aim being to introduce two geometric structures that a conformal 2-manifold might be equipped with, and to study the relationship between them. These structures are closely related to the projective and aane structures of Riemann surface theory. The rst structure can be viewed as a nonintegrable or nonholomorphic version of a complex projective structure, and will be called a MM obius structure. An integrable or at MM obius structure on a conformal 2-manifold induces a complex projective structure: the manifold possesses an atlas whose transition functions are complex MM obius transformations. However, contrary to common usage 9], the MM obius structures discussed herein are not necessarily integrable: they possess a curvature, analogous to the Cotton-York tensor of a conformal 3-manifold, whose vanishing is equivalent to integrability. MM obius structures are also diierent from real projective structures, in much the same way as conformal and real projective structures diier in higher dimensions. (In one dimension, MM obius and real projective structures do coincide and are always integrable.) The other topic of interest here is Einstein-Weyl geometry 3, 8, 14]. This is the geometry of a conformal manifold equipped with a compatible (or conformal) torsion free connection, such that the symmetric tracefree part of the Ricci tensor of this connection vanishes. These manifolds generalise Einstein manifolds in a natural way, and have been investigated in some detail recently (see 2, 6, 8] and references therein). In 11], Ped-ersen and Tod posed the problem of classifying compact two dimensional Einstein-Weyl manifolds|the possible geometries of compact three dimensional Einstein-Weyl mani-folds have been classiied (locally) by Tod 13]. However, the deenition just given of an Einstein-Weyl manifold is vacuous in the two dimensional case and Pedersen and Tod did not ooer an alternative deenition. One of the main goals of this paper is to give explicitly such a deenition and present a classiication of the compact orientable examples.

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