Non-holonomic and semi-holonomic frames in terms of Stiefel and Grassmann tangent bundles

Abstract

We prove that the bundles of non-holonomic and semi-holonomic second-order frames of a real or complex manifold M can be obtained as extensions of the bundle F 2( M) of second-order jets of (holomorphic) diffeomorphisms of ( K n,0) into M, where K= R or C . If dim K (M)=n and F(M) is the bundle of K -linear frames of M we will associate to the tangent bundle E=T( FM) two new bundles St n(E) and G n(E) with fibers of type the Stiefel manifold St n(V) and the Grassmann manifold G n(V) , respectively, where V= K n⊕ gl(n, K) . The natural projection of St n(E) onto G n(E) defines a GL(n, K) -principal bundle. We have found that the subset of G n(E) given by the horizontal n-planes is an open sub-bundle isomorphic to the bundle F ̂ 2(M) of semi-holonomic frames of second-order of M. Analogously, the subset of St n(E) given by the horizontal n-bases is an open sub-bundle which is isomorphic to the bundle F ̃ 2(M) of non-holonomic frames of second-order of M. Moreover the restriction of the former projection still defines a GL(n, K) -principal bundle. Since a linear connection is a horizontal distribution of n-planes invariant under the action of GL(n, K) it therefore determines a GL(n, K) -reduction of the bundle F ̂ 2(M) , in a bijective way. This is a new proof of a theorem of Libermann.