Abstract
In modern mathematics, the use of geometries with a maximum group of motions is of particular importance. There are many classifications of such geometries, one of which contains the geometry of a special extension of Euclidean space. This geometry belongs to the family of geometries with a degenerate Riemannian metric, but at the same time admits a group of motions of maximum dimension. This paper investigates the metric properties of the geometry of a special extension of Euclidean space. The concept of the length of a curve in such a geometry is introduced. The curve of the minimum length is found. It is proved that a segment in a horizontal hyperplane has the minimum length. The Christoffel symbols of the geometry of a special extension of Euclidean space are calculated.
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