Linear connections of holonomic and nonholonomic smooth manifolds

Yu. I. Shevchenko

Trudy Geometricheskogo Seminara, Trudy Geometricheskogo Seminara, Tome 23 (1997), pp. 175-186

Résumé

The terms “non-holonomic surface”, “non-holonomic manifold” and “non-holonomic space” are used in different senses by construction of structures generalizing in one way or another a surface in a homogeneous space or the space itself. On the one hand the notion of non-holonomic differentiable manifold generalizes a usual differentiable manifold which therefore we can naturally call holonomic one. On the other hand it generalizes Cartan's non-holonomic space, i.e. the space with fundamental-group connection, in particular, the affine connection space. Non-holonomic differentiable manifold was actually investigated by A. K. Rybnikov. This manifold is discovered on the way offered by M. A. Akivis and representing local approach to differentiable manifold. However, obtained results have global character, that was shown by G. F. Laptev and Yu. G. Lumiste in the different manners. Linear (in classical terminology — affine) connection on holonomic and non-holonomic differentiable manifolds have been studied which is considered as a group connection in the principle bundle of linear frames and have been defined by G. F. Laptev's way. It is proved, that torsion object is quasitensor on non-holonomic manifold, but curvature object forms quasitensor together with connection object only. They are tensors on holonomic manifold. In the non-holonomic case linear connection possesses torsion and curvature, whereas in the holonomic case, as it is known, one can consider connection without torsion or curvature. Rybnikov's geometric interpretation of the notion of connection is following: definition of linear connection (symmetric in holonomic case) on differentiable manifold is equivalent to equipment of manifold with field of subspaces, forming contact spaces of second order in direct sum with tangent spaces. Local geometric characteristic of such linear connection is given, which is treated inside of contact space by means of the projection of a neighbour tangent space on the initial tangent space in parallel to the equipping subspace.