\(tt^{*}\)-geometry on the tangent bundle of an almost complex manifold
A \(tt^*\)-bundle \((E,D,S)\) over an almost complex manifold \((M,J)\) is a real vector bundle \(E\to M\) endowed with a connection \(D\) and a section \(S\in\Gamma(T^* M\otimes \text{End\,}E)\) which satisfy the \(tt^*\)-equation: \(R^\theta= 0\) for all \(\theta\in\mathbb{R}\). Here \(R^\theta\) is the curvature of the connection \(D^\theta\) defined by: \[ D^\theta_X= D_X+ \cos\theta\cdot S_X+\sin\theta\cdot S_{JX}\text{ for all }X\in TM. \] A symplectic \(tt^*\)-bundle \((E,D,S,\omega)\) is a \(tt^*\)-bundle \((E,D,S)\) endowed with the structure of a symplectic vector bundle \((E,\omega)\) such that \(\omega\) is \(D\)-parallel and \(S\) is \(\omega\)-symmetric, i.e., for all \(p\in M\), \(\omega(S_X Y,Z)= \omega(Y, S_X Z)\) for all \(X,Y,Z\in T_p M\). The subject of this paper is \(tt^*\)-bundles \((TM,D,S)\) over an almost complex manifold \((M,J)\) which are induced by the one parameter family of connections \(\nabla^\theta= \exp(\theta J)\circ\nabla\circ\exp(-\theta J)\).
\(tt^{*}\)-geometry and \(tt^{*}\)-bundles
special complex and special Kähler manifolds
scientific article; zbMATH DE number 5125283
scientific article