Search
Submit an article RU
Differential Geometry of Manifolds

Current issue

Back to the list Download the article
Eliseeva N.A. Popov Yu.I.

Fields of fundamental and embracing geometric objects of a regular hyperband with central framing of a projective space

DOI
10.5922/0321-4796-2022-53-5
Pages
43-58
hyperstrip Cartan equipment regular hyperstrip torsion-curvature tensor quasitensor geometric object envelopment of a geomet­ric object projective connection

Abstract

The study of hyperbands and their generalizations in spaces with dif­ferent fundamental groups is of great interest in connection with numer­ous applications in mathematics and physics. In this paper, we study a special class of hyperbands, i. e., centrally equipped hyperbands. A hy­perband Hm (m ≥ 2) is said to be centrally rigged if the rigging lines in the normals of the 1st kind of the base surface pass through one (the center of the rigging).

The article gives a task of a centrally equipped hyperband in the 1st order frame. A sequence of fundamental geometric objects of a hyperstrip with central framing is constructed. An existence theorem for a hyperband with a central framing is proved. It is proved that a hyperstrip with central framing and framing in the sense of Cartan induces a projective connec­tion obtained by projection, where the projection center at each point is the Cartan plane. The spans of the components of the curvature-torsion tensor of the constructed connection are found.

Reference

1. Vagner, V. V.: The theory of the field of local hyperbands. Tr. Se­min. Vectorn. Tensorn. Anal., 8, 197—272 (1950).

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

3. Laptev, G. F.: Manifolds immersed in generalized spaces. Tr. 4th Vses. Math. Congress, 2, 226—236 (1961).

4. Popov, Yu. I.: Introduction of projective connections on the SH-dis­tribution of a projective space. IKBFU’s Vestnik. Physics, Math., and Techn., 1, 5—15 (2017).

5. Popov, Yu. I.: Introduction of connections on the hypersurface Ω(Δ). IKBFU’s Vestnik. Physics, Math., and Techn., 1, 18—24 (2018).

6. Popov, Yu. I.: General theory of regular hyperbands. Kaliningrad (1983).

7. Popov, Yu. I.: Contigious hyperquadrics of coequipped hyperbands sHm. DGMF. Kaliningrad. 50, 126—132 (2019).

8. Popov, Yu. I., Stolyarov, A. V.: Special classes of regular hyper­bands in a projective space. Kaliningrad (2011).

9. Popov, Yu. I.: Hyperstrip distributions of affine space. Kaliningrad (2021).

10. Popov, Yu. I.: On the fields of geometric objects of a D-framed hypersurface of a projective space. IKBFU’s Vestnik. Physics, Math., and Techn., 4, 16—23 (2017).

11. Stolyarov, A. V.: On fundamental objects of a regular hyperband. Izv. Vuzov. Math., 10, 97—99 (1979).

12. Finikov, S. P.: Cartan’s exterior form method in differential ge­ometry. Moscow (1948).

13. Cartan E.: Les espaces a connexion projective. Tr. Semin. Vek­torn. Tenzorn. Anal., 4, 147—159 (1937).