Symmetric bundles and representations of Lie triple systems

Wolfgang BERTRAM and Manon

Abstract

We dene symmetric bundles as vector bundles in the category of symmetric spaces; it is shown that this notion is the geometric analog of the one of a representation of a Lie triple system. A symmetric bundle has an underlying re ection space, and we investigate the corresponding forgetful functor both from the point of view of dierential geometry and from the point of view of representation theory. This functor is not injective, as is seen by constructing \unusual" symmetric bundle structures on the tangent bundles of certain symmetric spaces.