Curvature submodules for Grassmann structures with torsion

Abstract

A complete decomposition of the space R(Vp⊗q) of the curvature tensors over tensor product of vector spaces into simple modules under the action of the group G = GL(p, ℝ) ⊗ GL(q, ℝ) is given. We use these results to study geometry of manifolds with Grassmann structure and Grassmann manifolds endowed with a connection whose torsion is not zero. We show that Oscr M a manifold is an example of a manifold with Grassmann structure. Owing to this fact, we consider results of Miron, Atanasiu, Anastasiei, C;̌omić and others from representation theory point of view and connect them with some results of Alekseevsky, Cortes, and Devchand, as well as of Machida and Sato, and others. New examples of connections with torsion defined on four-dimensional Grassmann manifold are given. Symmetries of curvatures for half-flat connections are also investigated. We use algebraic results to reveal obstructions to the existence of corresponding connections. © 2006 World Scientific Publishing Company.