Almost Hermitian 6-Manifolds Revisited

Abstract

A Theorem of Kirichenko states that the torsion 3-form of the characteristic connection of a nearly Kähler manifold is parallel. On the other side, any almost hermitian manifold of type ${\mathrm{G}}_1$ admits a unique connection with totally skew symmetric torsion. In dimension six, we generalize Kirichenko's Theorem and we describe almost hermitian ${\mathrm{G}}_1$-manifolds with parallel torsion form. In particular, among them there are only two types of ${\mathcal{W}}_3$-manifolds with a non-abelian holonomy group, namely twistor spaces of 4-dimensional self-dual Einstein manifolds and the invariant hermitian structure on the Lie group $\mathrm{SL}(2, \C)$. Moreover, we classify all naturally reductive hermitian ${\mathcal{W}}_3$-manifolds with small isotropy group of the characteristic torsion.

26 pages, revised version

References in corpus (5)

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Cited by in corpus (26)

• The Srni lectures on non-integrable geometries with torsion
• The Ricci tensor of SU(3)-manifolds
• Nearly Kaehler Reduction
• SU(3)-instantons and $G_2,Spin(7)$-heterotic string solitons
• Spinorial description of $\mathrm{SU}(3)$- and $G_2$-manifolds
• Almost Hermitian Structures with Parallel Torsion
• Hermitian-Yang-Mills equations and pseudo-holomorphic bundles on nearly Kaehler and nearly Calabi-Yau twistor 6-manifolds
• G_2-structures with torsion from half-integrable nilmanifolds
• Metric connections with parallel skew-symmetric torsion
• Geometric structures of vectorial type
• On the Ricci tensor in type II B string theory
• Nearly integrable SO(3)-structures on 5-dimensional Lie groups
• $G_2$-Manifolds With Parallel Characteristic Torsion
• A method of deforming G-structures
• Spin(7)-manifolds with parallel torsion form
• The Killing spinor equation with higher order potentials
• A new $\frac{1}{2}$-Ricci type formula on the spinor bundle and applications
• Topological field theories of 2- and 3-forms in six dimensions
• A note on generalized Dirac eigenvalues for split holonomy and torsion
• Twistorial eigenvalue estimates for generalized Dirac operators with torsion
• Homogeneous almost complex structures in dimension 6 with semi-simple isotropy
• An integral formula for Riemannian $G$-structures with applications to almost Hermitian and almost contact structures
• On the metric structure of some non-Kähler complex threefolds
• Geometric Realizability of Covariant Derivative Kähler Tensors for almost Pseudo-Hermitian and almost Para-Hermitian Manifolds
• Einstein manifolds with skew torsion
• Eigenvalue estimates for Dirac operators with parallel characteristic torsion