Multidimensional three-webs that are formed from surfaces of different dimensions

M. A. Akivis ; V. V. Goldberg

Itogi Nauki i Tekhniki. Seriya Problemy Geometrii. Trudy Geometricheskogo Seminara, Trudy Geometricheskogo Seminara, Tome 4 (1973), pp. 179-204.

Résumé

Three-net formed by three families of surfaces of dimensions $p$, $q$, $r$ ($n=p+q$, $p\le q$) is studied on differentiable manifold $X_n$. Such a net is a differentiable $G$-structure on $X_n$. In the course of the examination three cases are naturally distinguished: $r\le p\le q$, $p\le r\le q$, $p\le q\le r$. In each of these cases the equations of structure are simplified by means of reducing the moving frame to canonical form, geometrical sense of the vanishing of some tensors associated with the three-net is elucidated, invariant normalization of some distribution of planes associated with the three-net is built, necessary and sufficient conditions of parallelizability of the three-net are pointed out.