Principal bundle structures among second order frame bundles

Abstract

Abstract. Using a model for the bundleF 2 M of semi-holonomic second order frames of a manifold M as an extension of the bundle F 2 M of holonomic second order frames of M , we introduce inF 2 M a principal bundle structure over F 2 M , the structure group being the additive group A 2 (n) of skew-symmetric bilinear maps from R 2 × R n into R n . The composition of the projection of that structure with the existing projection of the bundleF 2 M of non-holonomic second order frames of M overF 2 M provides a principal bundle structure inF 2 M over F 2 M . These results close an existing gap in the theory of second order frame bundles.

Lie groupoids and algebroids applied to the study of uniformity and homogeneity of Cosserat media

“…There are more principal bundles defined over the 1−jets of local isomorphisms j 1 e 1 ,Z Ψ on F M (holonomic and semiholonomic). To know about the relations between them see [11] Remark 2.6. Notice that there exists a canonical projectionπ 2 1 :…”