On types of non-integrable geometries

Abstract

We study the types of non-integrable $\mathrm{G}$-structures on Riemannian manifolds. In particular, geometric types admitting a connection with totally skew-symmetric torsion are characterized. 8-dimensional manifolds equipped with a $\Spin(7)$-structure play a special role. Any geometry of that type admits a unique connection with totally skew-symmetric torsion. Under weak conditions on the structure group we prove that this geometry is the only one with this property. Finally, we discuss the automorphism group of a Riemannian manifold with a fixed non-integrable $\mathrm{G}$-structure.

Latex2.09, 12 pages

References in corpus (4)

• G-Structures and Wrapped NS5-Branes
• Killing spinor equations in dimension 7 and geometry of integrable G_2-manifolds
• Cohomogeneity-one G2-structures
• Spin(9)-structures and connections with totally skew-symmetric torsion

Cited by in corpus (27)

• Non-Kaehler String Backgrounds and their Five Torsion Classes
• Scherk-Schwarz reduction of M-theory on G2-manifolds with fluxes
• Spinorial description of $\mathrm{SU}(3)$- and $G_2$-manifolds
• Almost Hermitian 6-Manifolds Revisited
• Prolongations of Lie algebras and applications
• Heterotic compactifications on nearly Kähler manifolds
• Hermitian-Yang-Mills equations and pseudo-holomorphic bundles on nearly Kaehler and nearly Calabi-Yau twistor 6-manifolds
• The Casimir operator of a metric connection with skew-symmetric torsion
• Geometric structures of vectorial type
• Compact supersymmetric solutions of the heterotic equations of motion in dimension 5
• Spin(9)-structures and connections with totally skew-symmetric torsion
• Connection with torsion, parallel spinors and geometry of Spin(7) manifolds
• Nearly integrable SO(3)-structures on 5-dimensional Lie groups
• A second look at gauged supergravities from fluxes in M-theory
• Sp(3) structures on 14-dimensional manifolds
• $G_2$-Manifolds With Parallel Characteristic Torsion
• Geometric structures associated with a simple Cartan 3-form
• The topological obstructions to the existence of an irreducible SO(3) structure on a five manifold
• Solvmanifolds with integrable and non-integrable G_2 structures
• A note on generalized Dirac eigenvalues for split holonomy and torsion
• An example of the geometry of a 5th-order ODE: the metric on the space of conics in ${\mathbb{CP}}^2$
• Special Riemannian geometries modeled on distinguished symmetric spaces
• Twistorial eigenvalue estimates for generalized Dirac operators with torsion
• Cones of G manifolds and Killing spinors with skew torsion
• SO(3)-Structures on 8-manifolds
• The geometry of the space of leaf closures of a transversely almost Kähler foliation
• On generalized quasi-Sasaki manifolds