The Grassmann-like manifold of centered planes when a surface is described by the centre

Abstract

We continue to study of the Grassmann-like manifold of -centered planes. A special case is considered when the center de­scribes an -dimensional surface . We will denote this mani­fold by . An analogue of the strong Norden normalization of the manifold is realized. It is proved that this normalization induces a connection in the bundle associated with the manifold . A geometric characteristic of this connection is given with the help of parallel displacements. In our research we use the Cartan method of external forms and the group-theoretical method of Laptev. These methods are used by many geometers and physicists. The Grassmann-like manifold is closely related to such a well-known and popular manifold as the Grassmann manifold. The Grassmann mani­fold is an example of a homogeneous space and forms an important fun­damental class of projective manifolds, and the projective space itself can be represented as a Grassmann manifold.

Keywords

• projective space
• the grassmann-like manifold
• surface
• connection
• parallel displacements