Non-holonomic and semi-holonomic frames in terms of Stiefel and Grassmann tangent bundles.
Let \(M\) be an \(n\)-dimensional real or complex manifold; the authors show that the bundles of non-holonomic and semi-holonomic second order frames of \(M\) can be obtained as extensions of the bundle of second order holonomic frames. If \(F(M)\) is the frame bundle of \(M\), two new bundles \(St_n(E)\) and \(G_n(E)\) are associated with the tangent bundle \(E=T(FM)\); their fibres are Stieffel and Grassmann type manifolds. The subset of \(St_n(E)\) (resp. \(G_n(E)\)) given by the horizontal \(n\)-bases (resp. \(n\)-planes) is an open subset isomorphic to the bundle of second order non-holonomic (resp. semi-holonomic) frames of \(M\). There is a natural projection \(St_n(E) \rightarrow G_n(E)\) which is a \(Gl(n,K)\)-fibre bundle (where \(K\) is the field of real or complex numbers); the same is true for the restriction to the open sub-bundles of horizontal \(n\)-bases and \(n\)-planes. Since a linear connection on \(M\) is a horizontal distribution of \(n\)-planes invariant under the action of the linear group \(Gl(n,K)\), their previous constructions allow the authors to give a new proof of a theorem by Liebermann, characterizing the linear connections on a real or complex manifold \(M\) as the set of \(Gl(n,K)\)-reductions of its bundle of second order semi-holonomic frames.
scientific article; zbMATH DE number 1974924
scientific article