The composition equipment for congruence of hypercentred planes

In n-dimensional projective space Pn a manifold Vnm , i. e., a congruence of hypercentered planes Pm , is considered. By a hypercentered planе Pm we mean m-dimensional plane with a (m – 1)-dimensional hyperplane Lm1 , distinguished in it. The first-order fundamental object  of the congruence is a pseudotensor. The principal fiber bundle Gr (Vnm) is associated with the congruence, r  n(n m1)  m2. . The base of the bundle is the manifold Vnm and a typical fiber is the stationarity subgroup Gr of a centered plane Pm . In principal fiber bundle a fundamental-group connection is given using the field of the object Г . The composition equipment for the congruence is set by means of a point lying in the plane and not belonging to its hypercenter and an (n – m – 1)-dimensional plane, which does not have common points with the hypercentered plane. The composition equipment is given by field of quasitensor  . It is proved that the composition equipment for the congruence Vnm of hypercentred m-planes Pm induces a fundamental-group connection with object Г in the principal bundle Gr (Vnm ) associated with the congruence. In proof, the envelopments Г  Г(, ) are built for the components of the connection object Г .

About an analogue of Neifeld’s connection on the space of centred planes with one-index basic-fibre forms

This research is realized by Cartan — Laptev method (with prolongations and scopes, moving frame and exterior forms). In this paper we consider a space П of centered m-planes (a space of all centered planes of the dimension m). This space is considered in the projective space n P . For the space П we have: dim П=n + (n – m)m. Principal fiber bundle is arised above it. The Lie group is a typical fiber of the principal fiber. This group acts in the tangent space to the П. Analogue of Neifeld’s connection with multivariate glueing is given in this fibering by Laptev — Lumiste way. The case when one-index forms are basic-fibre forms is considered. We realize an analogue of the Norden strong normalization of the space П by fields of the geometrical images: (n – m – 1)-plane which is not having the common points with a centered m-plane and (m – 1)-plane which is belonging to the m-plane and not passing through its centre. It is proved that the analog of the Norden strong normalization of the space of centered planes induces this connection.

Connections from trivializations

Let P be a principal fiber bundle with the basis M and with the structural group G. A trivialization of P is a section of P. It is proved that there exists only one gauge natural operator transforming trivializations of P into principal connections in P. All gauge natural operators transforming trivializations of P and torsion free classical linear connections on M into classical linear connections on P are completely described.

Topological invariance of the Hall conductance and quantization

It is shown that the Kubo equation for the Hall conductance can be expressed as an integral which implies quantization of the Hall conductance. The integral can be interpreted as the first Chern class of a [Formula: see text] principal fiber bundle on a two-dimensional torus. This accounts for the conductance given as an integer multiple of [Formula: see text]. The formalism can be extended to deduce the fractional conductivity as well.