tt∗-geometry on the tangent bundle of an almost complex manifold

The subject of this paper is tt∗-bundles (TM,D,S) over an almost complex manifold (M,J). Let ∇ be a flat connection on M. We characterize those tt∗-bundles with ∇=D+S which are induced by the one parameter family of connections ∇θ=exp(θJ)∘∇∘exp(−θJ) and obtain a uniqueness result for solutions where D is complex. A subclass of such solutions is flat nearly Kähler manifolds and special Kähler manifolds. Moreover, we study the case where these tt∗-bundles admit the structure of symplectic or metric tt∗-bundles. Finally, we generalize the notion of pluriharmonic maps to maps from almost complex manifolds (M,J) into pseudo-Riemannian manifolds and relate the above symplectic and metric tt∗-bundles to pluriharmonic maps from (M,J) into the pseudo-Riemannian symmetric spaces SO0(p,q)/U(p,q) and Sp(R2n)/U(p,q), respectively.