e10-7 by gegeshandong


									Russian Mathematics (Iz. VUZ)                                                                  Izvestiya VUZ. Matematika
Vol. 42, No. 7, pp.67{73, 1998                                                                              UDC 514.754
                              ON HYPERBANDS OF AFFINE SPACE
                               IN PETERSON CORRESPONDENCE
                                                       A.V. Chakmazyan

1. De nitions and results
   K.M. Peterson was the rst to consider a correspondence between surfaces in the three-
dimensional Euclidean space, more recently called the Peterson correspondence (see 1]). This
correspondence is characterized by the following property: Planes tangent to the surfaces at the
corresponding points are parallel. The Peterson correspondence was studied, e. g., in 2]{ 5]. Basi-
cally this correspondence is a ne rather than metric, therefore in 6] the Peterson correspondence
between surfaces was considered in an a ne space. On surfaces in a Peterson correspondence,
a ne-invariant a ne connections arise, which are used to investigate the local structure of the
   In the present article, following 6], we consider the Peterson correspondence between smooth
regular hyperbands in an a ne space.
    De nition 1. A smooth m-dimensional hyperband H in an n-dimensional a ne space A is a
smooth m-parametric manifold of hyperplane elements (x ), where the point x varies through an
m-dimensional surface M , and the hyperplane = (x) is tangent to M at x 2 M (see 7]).
   The surface M is called the basic surface of the hyperband H , and the hyperplanes (x) are
called the principal tangent hyperplanes of H . If at each point x 2 M the (n ; m ; 1)-dimensional
characteristic plane (x) of the family of principal tangent hyperplanes of H does not contain
directions tangent to M at x, then H is said to be regular (see 7]). An intrinsic invariant framing
of regular hyperband in a ne space was considered by Yu.I. Popov in 8].
   For a hyperband H the set of tangent planes of basic surface constitutes the tangent centroa ne
bundle T (M ), and the set of characteristic planes of principal tangent hyperplanes of H forms a
characteristic bundle (H ).
    De nition 2. Two smooth regular hyperbands H and H in an a ne space A are said to be in
a Peterson pre-correspondence if a one-to-one correspondence between their points exists such that
the principal tangent planes at the corresponding points of hyperbands are parallel.
   Lemma. Let two smooth regular hyperbands H and H in a ne space be in a Peterson pre-
correspondence. Then the characteristic planes (x) and (y) of the principal tangent hyperplanes
of H and H , which are taken at the corresponding points, are parallel.

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