On types of non-integrable geometries

2002/05/14 by Thomas Friedrich, Friedrich, Thomas · 2 citations
Mathematics · Medicine · Physics and Astronomy · #53C25 #81T30 #Advanced Differential Geometry Research #Advanced Neuroimaging Techniques and Applications #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Geometric Analysis and Curvature Flows #High Energy Physics - Theory (hep-th) #hep-th #math.DG #msc:53C25 #msc:81T30

paper · pdf · doi:10.48550/arxiv.math/0205149

Latex2.09, 12 pages

arxiv created 2002/05/14 · openalex publication_date 2002/05/14 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the types of non-integrable 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 G-structure.

Cited by

• Parallel torsion and G₂, Spin(7) instantons
• The topological obstructions to the existence of an irreducible SO(3) structure on a five manifold

Related

• The Srni lectures on non-integrable geometries with torsion
• Intrinsic torsion classes of Riemannian structures
• \mathrm G₂-structures with parallel skew-symmetric torsion
• Frölicher-Nijenhuis bracket and geometry of G₂-and \rm Spin(7)-manifolds
• G₂-Manifolds With Parallel Characteristic Torsion