Differential Geometry of Manifolds

2023 №54(1)

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The deformation pseudotensor of connections in cocongruence K (n - m)m



The Grassmann manifold  is the set of all -dimensional planes of an -dimensional projective space, with


One of the submanifolds of the Grassmann manifold is a complex of -planes if the dimension of the complex exceeds the difference .

We continue to study the cocongruence of -dimensional planes us­ing the Cartan — Laptev method. In an -dimensional projective space, the cocongruence of -dimensional planes can be given by the following equations .

Compositional equipment of a given cocongruence by fields of

()-planes :

and points

allows one to define connections of three types in the associated bundle, and one of the three connections is average with respect to the other two. The deformation of these connections is considered and it is shown that the object of deformation is a pseudotensor.

We introduce the deformation object  of the connection of the sec­ond type with respect to the connection of the first type. The deformation of the connection of the third type with respect to the connection of the first type is , and the deformation of the connection of the third type with respect to the connection of the second type is .

In the present paper, we use the method of continuations and cover­ages of G. F. Laptev with assignment of connections in the principal bun­dle.


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