**Eteri Alshibaia**

## On a Normal of the Distribution of
Hyperplane Elements in an Affine Space

**Abstract:**

Using the Cartan-Laptev invariant analytic method, an invariant affine normal
$(E)$ is constructed, which is intrinsically connected with the distribution of
hyperplane elements in the $(n+1)$-dimensional affine space. For the normal
$(E)$ we define a second kind normal and an invariant $(n-1)$-dimensional plane
lying in the plane of the element and not passing through the center and
corresponding to this normal in the Bompiani-Pantazi projectivity. An invariant
point of intersection of the two-dimensional plane passing through the normals
$(L)$ and $(E)$ with the second kind normal is found. The construction is
carried out without assuming that the nonholonomy tensor is different from zero.
Hence both nonholonomic and holonomic distributions are framed by the
constructed normal.

**Keywords:**

Fibre space, hyperplane element, holonomic and nonholonomic distributions,
differential continuation, affine normal, fundamental object.

**MSC 2000:** 53A15