We define 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 reflection space, and we investigate the corresponding forgetful functor both from the point of view of differential 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.1 A word of warning: in the literature, especially on Jordan algebras, there is some confusion in terminology; the notion of general representation differs very much from the idea of a representation to be a homomorphism "into some matrix realization". Unfortunately, the word "representation" is also used in this second sense for Jordan algebras (cf., e.g., [FK94]) and for symmetric spaces ([Be00, I.5]); we suggest to replace this by the term "specialization", in the sense of "homomorphism into a special (i.e., matrix or operator) object".

We define 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 reflection space, and we investigate the corresponding forgetful functor both from the point of view of differential 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.2000 MSC: 17A01, 17B10, 53C35.