Differential Geometry of Manifolds

2020 №51

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Fields of geometric objects associated with compiled hyperplane -distribution in affine space



A compiled hyperplane distribution  is considered in an n-dimensional projective space . We will briefly call it a -distribution. Note that the plane L(A) is the distribution characteristic obtained by displacement in the center belonging to the L-subbundle. The following results were obtained:
a) The existence theorem is proved: -distribution exists with arbitrary (3n – 5) functions of n arguments.
b) A focal manifold  is constructed in the normal plane  of the 1st kind of L-subbundle. It was obtained by shifting the cen­ter A along the curves belonging to the L-distribution. A focal manifold  is also given, which is an analog of the Koenigs plane for the distribution pair (L, L).
c) It is shown that a framed -distribution in the 1st kind normal field of H-distribution induces tangent    and  normal bundles.
d) Six connection theorems induced by a framed -distri­bu­tion in these bundles are proved.
In each of the bundles ,  the framed -distribution induces an intrin­sic torsion-free affine connection in the tangent bundle and a centro-affine connection in the corresponding normal bundle.
e) In each of the bundles (d) in the differential neighborhood of the 2nd order, the covers of 2-forms of curvature and curvature tensors of the corresponding connections are constructed.


1. Popov Yu. I.: Normalization of main structural subbundles -distributions of affine space. IKBFU's Vestnik. Ser. Physics, Mathematics, and Technology, 3, 5—14 (2018).

2. Laptev, G. F., Ostianu, N. M.: Distributions of m-dimensional line ele­ments in a space with projective connection. Tr. Geom. Sem., 3, 49—94 (1971).

3. Popov, Yu. I.: Three-part distribution of projective space. DGMF. Kaliningrad. 18, 65—86 (1987).

4. Norden, A. P.: Normalization theory and vector bundles. Tr. Ge­om. Sem. Kazan Univ. 9, 68—76 (1976).

5. Laptev, G. F.: Differential geometry of imbedded manifolds. Group theoretical method of differential geometric investigations. Tr. Mosk. Mat. Obs., 2, 275—382 (1953).

6. Stoljarov, A. V.: The projective differential geometry of a regular hyperband distribution of m-dimensional line elements. Itogi nauki i tekhn. Ser. Probl. Geom., 7, 117—151 (1975).

7. Chakmazyan, A. V.: Normal connection in the geometry of framed submanifolds. Yerevan (1999).