2015/03/04 by Izu Vaisman, Vaisman, Izu
Mathematics · #53C15 #Differential Geometry (math.DG) #FOS: Mathematics #math.DG #msc:53C15
paper · pdf · doi:10.48550/arxiv.1503.01251
LaTeX, 22 pages. The second version includes an additional material
arxiv created 2015/04/20 · arxiv updated 2015/04/21
We define a generalized almost para-Hermitian structure to be a commuting pair (F,J) of a generalized almost para-complex structure and a generalized almost complex structure with an adequate non-degeneracy condition. If the two structures are integrable the pair is called a generalized para-Kähler structure. This class of structures contains both the classical para-Kähler structure and the classical Kähler structure. We show that a generalized almost para-Hermitian structure is equivalent to a triple (γ,ψ,F), where γ is a (pseudo) Riemannian metric, ψ is a 2-form and F is a complex (1,1)-tensor field such that F2=Id,γ(FX,Y)+γ(X,FY)=0. We deduce integrability conditions similar to those of the generalized Kähler structures and give several examples of generalized para-Kähler manifolds. We discuss submanifolds that bear induced para-Kähler structures and, on the other hand, we define a reduction process of para-Kähler structures.