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Descripción
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| An affi?ne hypersurface (AH) structure is a pair comprising a conformal structure and a projective structure such that for any torsion-free connection representing the projective structure the completely trace-free part of the covariant derivative of any metric representing the conformal structure is completely symmetric. AH structures simultaneously generalize Weyl structures and abstract the geometric structure determined on a nondegenerate co-oriented hypersurface in flat affi?ne space by its second fundamental form together with either the projective structure induced by the a?ne normal or that induced by the conormal Gauss map. I will describe some notions of Einstein equations for AH structures which for Weyl structures specialize to the usual Einstein Weyl equations and such that the AH structure induced on a nondegenerate co-oriented a?ne hypersurface is Einstein if and only if the hypersurface is an affine sphere. In particular, by a theorem of Cheng-Yau, a properly convex flat projective structure admits a metric with which it generates an Einstein AH structure. There are other examples that arise from neither Weyl structures nor a?ne spheres. I will explain that, although it is much more general than conformal geometry, many of the structures and problems arising in conformal geometry have or should have extensions to the context of AH structures. | |
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Internacional
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Si |
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ISSN o ISBN
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Entidad relacionada
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Nacionalidad Entidad
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ESPAÑA |
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Lugar del congreso
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Centre de Recerca Matematica (Bellaterra, España) |