Half-dimensional immersions in para-Kähler manifolds
Résumé
We show that every in a certain sense non-degenerated Lagrangian immersion in a para-Kähler 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-Kähler manifold. We derive this equivalence from a similar, but more general one, relating non-degenerated half-dimensional immersions in para-
Kähler manifolds to dual pairs of what we call pre-Codazzi structures. We specialize this equivalence in two cases. Firstly, we show that every projectively
flat manifold carries a natural pre-Codazzi structure, and can be, at least locally, represented as a half-dimensional immersion in a special para-
Kähler manifold, which we call the cross-ratio manifold. Secondly, we show that manifolds carrying pre-Codazzi structures with flat connections are represented by half-dimensional immersions in the flat para-Kähler space. Our results have applications in affine differential geometry. Namely, centro-affine
geometry can be seen as the geometry of Lagrangian immersions in the crossratio manifold, while the geometry of graph immersions is equivalent to the
geometry of Lagrangian immersions in the flat para-Kähler 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.