2013 Issue №10

In n-dimensional space of affine connection An,n with Cartan’s structure equations Ricci’s and Bianchi’s identities were received. Their invariance has been shown. After prolongation of the structure equations using Laptev’s lemma semiholonomical, holonomical and trivial manifolds are defined. The Ricci’s identities allowed us to prove semiholonomicity of the space An,n. This semiholonomicity preserves in the space without torsion A’n,n and in the space without curvature ‘An,n , besides the locally affine space Ά’n,n is trivial. Tensor of non-holonomicity of the space An,n is introduced. Vanishing of this tensor makes the space holonomic, H n n A . Also curvature tensor of associated space of affine connection without torsion A’n,n was introduced. It’s vanishing characterizes trivial space of affine connection, Tn n A .

Regular hyperband distribution of an affine space (H-distribution) is given and its existence theorem is proved. Conditions of holonomic H-distribution is determined and Bompiani—Pantazi bijection between the first kind normal and the second one for associated with H-distribution H-, Λ- and L-subbundle is introduced.

In projective space the Grassmann-like manifold of centered planes is considered. Parallel displacements of directions on this manifold are studied.

The paper is concerned with m-dimensional surface in n-dimensional projective space. In studying fundamental-group connection Bianchi identities are found. It is proved that alternated covariant derivatives for the components of the first type connection object are equal to the corresponding components of the curvature tensor, and the ones of the third type vanish.

In multidimensional projective space a family of hyperplane elements with envelope surface of centers is considered. The problem of construction of invariant clothing intrinsically attached to such a family intrinsically is set. This problem is solved in a special case characterized by vanishing of a certain tensor. The solution is based on the method of moving frames and calculation of exterior differential forms of E. Cartan.