tt*-bundles in para-complex geometry, special para-Kähler manifolds and para-pluriharmonic maps
Résumé
We introduce the notion of a para-tt*-bundle, the generalization of a tt*-bundle in para-complex geometry. The main result is the definition of a map $\Phi$ from the space of metric para-tt*-bundles of rank r over a para-complex manifold M to the space of para-pluriharmonic maps from M to $GL(r)/O(p,q)$ where (p,q) is the signature of the metric and the description of the image of this map $\Phi.$ Then we recall and prove some results known in special complex and special Kähler geometry in the setting of para-complex geometry, which we use in the sequel to give a simple characterization of the tangent bundle of a special para-complex and special para-Kähler manifold as a particular type of tt*-bundles. For the case of a special para-Kähler manifold it is shown that the para-pluriharmonic map coincides with the dual Gauss map, which is a para-holomorphic map into the symmetric space ${\rm Sp}(\bR^{2n})/{\rm U}^{\pi}(C^n) \subset {\rm SL}(2n)/{\rm SO}(n,n)\subset {\rm GL}(2n)/{\rm O}(n,n).$
Origine : Fichiers produits par l'(les) auteur(s)
Loading...