Non-holonomic and semi-holonomic frames in terms of Stiefel and Grassmann tangent bundles
✍ Scribed by A. Martı́n Méndez; J.F. Torres Lopera
- Publisher
- Elsevier Science
- Year
- 2003
- Tongue
- English
- Weight
- 195 KB
- Volume
- 47
- Category
- Article
- ISSN
- 0393-0440
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✦ Synopsis
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
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.