Lie triple system bundles and algebroids

The aim of this article is to introduce the notion of a Lie triple system bundle which arises naturally from the tangent bundle of a locally symmetric space. We introduce suitable notion of representation and define cohomology groups of a Lie triple system bundle with coefficients in a given representation. We also introduce Generalized Reynolds operators and its variants on a Lie triple system bundle. We show that the cohomology groups introduced in the present paper turns out to be the deformation cohomology to develop formal deformation theory of Lie triple system bundles. The notion of infinitesimal deformation, rigidity, and integrability of an infinitesimal deformation are introduced in the process. Finally, a plausible definition of a Lie triple system algebroid is discussed. It turns out that a Lie triple system algebroid under certain conditions gives rise to a Lie triple system bundle.