Framed hyperstrip affine space distribution
Abstract
A rigged hyperstrip distribution is a special class of hyperstrips. The study of hyperstrips and their generalizations in spaces with various fundamental groups is of great interest due to numerous applications in mathematics and physics. A special place is occupied by regular hyperstrips, for which the characteristic planes of families of principal tangent hyperplanes do not contain directions tangent to the base surface of the hyperstrip. In this work, we use E. Cartan’s method of external differential forms and the group-theoretic method of G. F. Laptev.
In affine space, a hyperstrip distribution is considered, which at each point of the base surface is equipped with a tangent plane and a conjugate tangent line. The specification of the studied hyperstrip distribution in an affine space with respect to a 1st order reference and an existence theorem are given. The fields of affine normals of the 1st kind for Blaschke and Transon are constructed and the conditions for their coincidence are found. The definition of normal affine connection and normal centroaffine connection on the studied framed hyperstrip distribution is given.
Reference
1. Akivis, M. A., Rosenfeld, B. A.: Eli Cartan (1869—1951). Moscow (2014).
2. Vagner, V. V.: The theory of the field of local hyperstrips. Tr. Semin. Vectorn. Tensorn. Anal., 8, 197—272 (1950).
3. Eliseeva, N. A., Popov, Yu. I.: Hyperband distribution equipped with a field of conjugate planes. DGMF, 54 (1), 78—91 (2023).
4. Laptev, G. F.: Differential geometry of imbedded manifolds. Group theoretical method of differential geometric investigations. Tr. Mosk. Mat. Obs., 2, 275—382 (1953).
5. Ostianu, N. M., Ryzhkov, V. V., Shveikin P. I.: Outline of scientific research by German Fedorovich Laptev. Tr. Geom. Sem., 4, 7—70 (1975).
6. Popov, Yu. I.: Hyperband distribution of an affine space. Itogi Nauki i Tekhn. 203, 84—99 (2021).
7. Popov, Yu. I.: Hyperband distributions of affine space. Kaliningrad (2021).
8. Popov, Yu. I.: Introduction to the theory of a regular hyper-band distribution of an affine space. IKBFU’s Vestnik, 10, 49—56 (2013).
9. Popov, Yu. I.: Special classes of hyperband distribution of an affine space. Kaliningrad (2021).
10. Stolyarov, A. V.: Projective-differential geometry of a regular hyperstrip distribution of m-dimensional linear elements. Problems of Geom., 7, 117—151 (1975).
11. Finikov, S. P.: Cartan’s exterior form method in differential geometry. Moscow (1948).
12. Chakmazyan, A. V.: Normal connection in the geometry of submanifolds. Yerevan (1990).
13. An-Min, L., Simon, U., Guosong, Zh., Zejun, H.: Global Affine. Differential Geometry of Hypersurfaces, De Gruyter (Expositions in Mathematics, 11) (2015).
14. Akivis, M. A.: Selected Papers. Heldermann (2008).
15. Ivey, Th. A., Landsberg, J. M.: Cartan for Beginners: Differential Geometry Via Moving Frames and Exterior Differential Systems. Amer. Math. Soc. (Graduate Studies in Mathematics, 61) (2003).