\(tt^{*}\)-geometry on the tangent bundle of an almost complex manifold
type: Item
instance of: scholarly article
title: \(tt^{*}\)-geometry on the tangent bundle of an almost complex manifold
zbMATH Open document ID: 1121.32013
author: Lars Schäfer
published in: Journal of Geometry and Physics
publication date: 2007-02-13T00:00:00Z
review text: 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 \(\nabla^\theta= \exp(\theta J)\circ\nabla\circ\exp(-\theta J)\).
Mathematics Subject Classification ID: 32Q60
zbMATH DE Number: 5125283
zbMATH Keywords: \(tt^{*}\)-geometry and \(tt^{*}\)-bundles
zbMATH Keywords: special complex and special Kähler manifolds
zbMATH Keywords: nearly Kähler manifolds
zbMATH Keywords: pluriharmonic maps
zbMATH Keywords: pseudo-Riemannian symmetric spaces
zbMATH Keywords: connection
zbMATH Keywords: symplectic vector bundle
reviewed by: S. Ivashkovich
MaRDI profile type: Publication
full work available at URL: j.geomphys.2006.08.004
OpenAlex ID: W2037619149
cites work: Special complex manifolds
cites work: A decomposition of a holomorphic vector bundle with connection and its applications to complex affine immersions
cites work: Q5465259
cites work: Flat nearly Kähler manifolds
cites work: Geometry and integrability of topological-antitopological fusion
cites work: Associated families of the pluriharmonic maps and isotropy
cites work: Special Kähler manifolds
cites work: Parallel spinors and connections with skew-symmetric torsion in string theory.
cites work: Q5565587
cites work: Corrigenda to: Introduction to Symplectic Topology
cites work: On nearly-Kähler geometry
cites work: Nearly Kähler geometry and Riemannian foliations.
cites work: \(tt^*\)-geometry and pluriharmonic maps
cites work: Harmonic bundles, topological-antitopological fusion and the related pluriharmonic maps
DOI: 10.1016/J.GEOMPHYS.2006.08.004
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description: scientific article; zbMATH DE number 5125283