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

Lars Schaefer (1, 2)

1 IECN - Institut Élie Cartan de Nancy
2 Rheinische Friedrich-Wilhelms-Universität Bonn

The subject of this paper are tt*-bundles (TM,D,S) over an almost complex manifold (M,J). Let $\nabla$ be a flat connection on M. We characterize those tt*-bundles with $\nabla=D+S$ which are induced by the one parameter family of connections $\nabla^{\theta} = \exp{(\theta J)} \circ \nabla \circ \exp{(-\theta J)}$ and obtain a uniqueness result for solutions where $D$ is complex. A subclass of such solutions are 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 $SO_0(p,q)/U(p,q)$ and $\mathrm{Sp}(\bR^{2n})/U(p,q),$ respectively.

Keywords: tt*-geometry and tt*-bundles; special complex and special Kähler manifolds; Nearly Kähler manifolds; Pluriharmonic maps; Pseudo-Riemannian symmertic spaces

Domains: Differential Geometry [math.DG]

Lars Schaefer. tt*-geometry on the tangent bundle of an almost complex manifold. Journal of Geometry and Physics, 2007, 57, pp.999-1014. ⟨hal-00143288⟩