2026 №57 (1)

On a subclass of ellipsoid complexes that admits a non-integral representation

DOI

10.5922/0321-4796-2026-57-1-5

Pages

64-75

Abstract

Using the group-theoretic method of differential geometric re­search by G. F. Laptev, as well as the methodology of scientific re­search on differential geometry by V. S. Malakhovsky, the study of complexes (three-parameter families) of ellipsoids with special pro­perties of differentiable maps associated with them in a three-di­mensional affine space. The complex of ellipsoids is considered as an image of the corresponding mapping. A subclass of previous­ly studied ellipsoid complexes is investigated when the difference indicatrix of the second and third coordinate vectors is a straight line parallel to the first coordinate vector. Interesting geo­metric properties of the studied manifold are found, which make it possi­ble to construct it, that is, to find its integral representation.

Reference

1. Malakhovsky, V. S.: Differential geometry of manifolds of figures and pairs of figures in a homogeneous space. Tr. Geom. Sem., 2, 179—206 (1969).

2. Rosenfeld, B. A.: Multidimensional spaces. Moscow (1966).

3. Laptev, G. F.: Towards an invariant analytical theory of differentia­ble mappings. Tr. Geom. Sem., 6, 37—42 (1974).

4. Malakhovsky, V. S.: Differential geometry of manifolds of quadratic elements. Tomsk (PhD thesis) (1964).

5. Malakhovsky, V. S.: Some problems of differential geometry of ma­nifolds of shapes. DGMF, 5, 64—84 (1974).

6. Malakhovsky, V. S.: Introduction to the theory of external forms. Kaliningrad (1978).

7. Malakhovsky, V. S.: A Short Course in Differential Geometry. Kali­nin­grad (2010).

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

9. Cartan, E.: Theory of finite continuous groups and differential ge­ometry developed by the method of the Moving Frame. Moscow (1963).

10. Kretov, M. V.: Complexes of ellipsoids with special properties of differentiable maps associated with them. DGMF, 17, 51—57 (1986).

11. Kretov, M. V.: On special subclasses of differentiable maps asso­ci­ated with complexes of central non-degenerate hyperquadrics. DGMF, 15, 49—54 (1984).

12. Kretov, M. V.: On a complex of central quadrics with a degenera­ting center variety. Materials of the VII All-Union Conference on Modern Problems of Geometry. Minsk, 99 (1979).

13. Kretov, M. V.: Differentiable mappings associated with hyperqu­ad­ric manifolds. Materials of the International Conference on Geometry and Applications. Smolyan, 23 (1986).

14. Malakhovski, V. S., Makhorkin, V. V.: Differential geometry of va­rieties of hyperquadrics in -dimensional projective space. Tr. Geom. Sem., 6, 113—134 (1974).

15. Kretov, M. V.: On some subclasses of ellipsoid complexes in affi­ne space. Kaliningrad State University. Kaliningrad. Dep. VINITI 17.11.1981, No. 5272-81 (1981).

16. Kretov, M. V.: Geometric model of a three-parameter family of el­lipsoids. IKBFU’s Vestnik. Physics, Math., 4, 163—167 (2014).