A matrix representation of a~semiholonomic differential group, and the structure equations of the $p$-coframe bundle

Ü. G. Lumiste

Itogi Nauki i Tekhniki. Seriya Problemy Geometrii. Trudy Geometricheskogo Seminara, Trudy Geometricheskogo Seminara, Tome 5 (1974), pp. 239-257

Résumé

In the development of the theory of differential prolongations of smooth manifolds there are two directions: nowadays ?one in based on Ehresmann's theory or jets and another on E. Cartan's calculus of exterior forms and G. F. Laptev's structure equations. Relations between the two directions are investigated in [3], [13], [20] and [22]. This article presents for these relations a new interpretation, given first in [10]. The well-known concept of principal fibre bundle of semi-holonomic $р$-согерег ${}^*\overline H{}^p(M)$ of a differentiable manifold $M$ is here introduced in the way giving immediately the explicite representation of the structure group $\overline L_n^p$ of the bundle ${}^*\overline H{}^p(M)$. The group $\overline L_n^p$, the so-called semi-holonomic differential group, is obtained as a Lie subgroup of the $GL(N,R)$, where $N=n+n^2+\dots+n^p$. The Maurer-Cartan structure equations for $\overline L_n^p$ follow now directly from the well-known equations for $GL(N,R)$. They make it possible to use the Laptev structure equations for general principal fibre bundle global character of which is shown in [8], [7]. As a result we get the structure equations of the principal fibre bundle ${}^*\overline H{}^p(M)$, first given in [8]. In addition, some algebraic properties of $\overline L_n^p$ are investigated (cf. [11], [16]).