Holonomy, Geometry and Topology of Manifolds with Grassmann Structure

Abstract

The main goal is to present how geometry and algebra intertwine in studying some topics. Let ℜ(G) be the vector bundle of all curvature tensors corresponding to torsion-free connections on a smooth manifold M whose holonomy group is G. Here we present an overview on ℜ(G) from the representation theory point of view. We pay attention especially to manifolds endowed with Grassmann structure. Some examples of this type of manifolds are given as well as some topological obstructions for the existence of this structure. It is confirmed that many geometrical properties of manifolds with Grassmann structure are related to some symmetry properties of a curvature, which belongs to some simple G-modules of ℜ(G). A manifold may be endowed with various types of connections. The most interesting are those whose curvatures belong either to some simple submodules or to some of their direct sums as they define some special geometry of manifolds with Grassmann structure. Among these types of connections we point out: half-flat Grassmann connections, connections corresponding to some normalization, projectively equivalent connections, connections with symmetric or skew-symmetric Ricci tensor, etc. Having in mind a complete decomposition of ℜ(G). into simple submodules we can also reveal some new curvature invariants corresponding to some transformations of manifolds with Grassmann structure. Some relations between projective structures and corresponding reductions of a structure group are presented. We also give an overview of relations between projective geometry of manifolds with a special type of Grassmann structure and Riccati type equations. Various examples of previously mentioned facts are also provided.