Principal bundle structures among second order frame bundles

Using a model for the bundle Fˆ2M of semi-holonomic second order frames of a manifold M   as an extension of the bundle F2MF2M of holonomic second order frames of M  , we introduce in Fˆ2M a principal bundle structure over F2MF2M, the structure group being the additive group A2(n)A2(n) of skew-symmetric bilinear maps from Rn×RnRn×Rn into RnRn. The composition of the projection of that structure with the existing projection of the bundle F˜2M of non-holonomic second order frames of M   over Fˆ2M provides a principal bundle structure in F˜2M over F2MF2M. These results close an existing gap in the theory of second order frame bundles.