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- W2588524043 abstract "We show that every in a certain sense non-degenerated Lagrangian immersion in a para-Kahler manifold naturally carries a dual pair of Codazzi structures. On the other hand, every manifold carrying a dual pair of Codazzi structures can be represented as a non-degenerated Lagrangian submanifold of a para-Kahler manifold. We derive this equivalence from a similar, but more general one, relating non-degenerated half-dimensional immersions in para- Kahler manifolds to dual pairs of what we call pre-Codazzi structures. We specialize this equivalence in two cases. Firstly, we show that every projec- tively flat manifold carries a natural pre-Codazzi structure, and can be, at least locally, represented as a half-dimensional immersion in a special para- Kahler manifold, which we call the cross-ratio manifold. Secondly, we show that manifolds carrying pre-Codazzi structures with flat connections are rep- resented by half-dimensional immersions in the flat para-Kahler space. Our results have applications in affine differential geometry. Namely, centro-affine geometry can be seen as the geometry of Lagrangian immersions in the cross- ratio manifold, while the geometry of graph immersions is equivalent to the geometry of Lagrangian immersions in the flat para-Kahler space. We also obtain a natural duality relation between projectively flat connections on a manifold, extending the duality induced by the conormal map of centro-affine immersions to connections which are not equiaffine." @default.
- W2588524043 created "2017-02-24" @default.
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- W2588524043 date "2011-04-01" @default.
- W2588524043 modified "2023-10-14" @default.
- W2588524043 title "Half-dimensional immersions in para-Kähler manifolds" @default.
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