Skip to main content
Log in

Normal connection in the geometry of normalized submanifolds of affine space

  • Published:
Journal of Soviet Mathematics Aims and scope Submit manuscript

Abstract

A survey is given of results concerning the study of the structure of a normalized subraanifold of affine space with the help of a normal connection.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

Literature cited

  1. M. A. Akivis, “Focal forms of a surface of rank r,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 1, 9–19 (1957).

    Google Scholar 

  2. M. A. Akivis, “Multidimensional surfaces supporting a net of conjugate lines,” Dokl. Akad. Nauk SSSR,139, No. 6, 1279–1282 (1961).

    Google Scholar 

  3. M. A. Akivis, “Voss normals of a surface supporting a net of conjugate lines,” Mat. Shorn.,58, No. 2, 695–706 (1962).

    Google Scholar 

  4. M. A. Akivis and A. V. Chakmazyan, “Normalized submanifolds of affine space admitting a parallel normal vector field,” Dokl. Akad. Nauk ArmSSR,60, No. 3, 137–143 (1975).

    Google Scholar 

  5. E. D. Alshibaya, “Differential geometry of a hypersurface in a multidimensional affine space,” Tr. Tbilis. Univ.,129, 319–341 (1968).

    Google Scholar 

  6. L. S. Atanasyan, “Normalized manifolds of special form in a multidimensional affine space,” Tr. Sem. po Vektorn. i Tenzorn. Analizu, No. 9, 351–410 (1952).

    Google Scholar 

  7. V. T. Bazylev, “Multidimensional nets and their transformations,” Itogi Nauki Tekh., Ser. Geom., 1963, 138–164 (1965).

    Google Scholar 

  8. E. Cartan, Riemannian Geometry in an Orthonormal Frame [Russian translation], MGU, Moscow (1960).

    Google Scholar 

  9. G. F. Laptev, “Invariant normalization of a surface in a space with affine connection,” Dokl. Akad. Nauk SSSR,126, No. 3, 490–493 (1959).

    Google Scholar 

  10. Yu. G. Lumiste, “Differential geometry of submanifolds,” Itogi Nauki Tekh., Ser, Alg. Topol. Geom.,13, 273–340 (1975).

    Google Scholar 

  11. Yu. G. Lumiste and A. V. Chakmazyan, “Normal connection and submanifold with parallel normal fields in a space of constant curvature,” Itogi Nauki Tekh. Ser. Probl. Geom.,12, 3–30 (1980).

    Google Scholar 

  12. K. Nomidzu, Lie Group and Differential Geometry [Russian translation], Moscow (1960).

  13. A. V. Norden, Spaces with Affine Connection [in Russian], 2nd edition, Nauka, Moscow (1976).

    Google Scholar 

  14. N. M. Ostianu, V. V. Ryzhkov, and P. I. Shveikin, “Sketch of the research of German Fedorovich Laptev,” Tr. Geom. Semin. Vses. Inst. Nauchn. Tekhn. Inform.,4, 7–70 (1973).

    Google Scholar 

  15. D. I. Perepelkin, “Curvature and normal spaces of a manifold Vm in Rn,” Mat. Sborn.,42, No. 1, 81–120 (1935).

    Google Scholar 

  16. D. I. Perepelkin, “Parallel submanifolds in Euclidean (or Riemannian) spaces,” Dokl. Akad. Nauk SSSR,1, 593–598 (1935).

    Google Scholar 

  17. A. V. Chakmazyan, “Submanifold of projective space with parallel subbundle of the normal bundle,” in: Abstracts of Reports. National Geometry Conference “150 Years of Non-Euclidean Geometry” [in Russian], Kazan' (1976), p. 209.

  18. A. V. Chakmazyan, “Normalized submanifolds of affine space with flat normal affine connection,” in: Differential Geometry [in Russian], Kalinin (1977), pp. 120–129.

  19. A, V. Chakmazyan, “Connection in normal bundles of a normal submanifold Vm in Pn,” Itogi Nauki Tekh. Ser. Probl. Geom.,10, 55–75 (1978).

    Google Scholar 

  20. A. V. Chakmazyan, “Normalized submanifolds with flat normal connection in projective space,” Mat. Zametki,33, No. 2, 281–288 (1983).

    Google Scholar 

  21. A. V. Chakmazyan, “Affine geometry of a normalized submanifold with parallel field of normal P-directions,” Uch. Zap. Tart. Univ., No. 665, 81–89 (1984).

    Google Scholar 

  22. A. V. Chakmazyan, “Normalizations with flat normal connection for submanifolds of affine space,” Izv. Vuzov. Matematika,1, 74–79 (1987).

    Google Scholar 

  23. P. A. Shirokov and A. P. Shirokov, Affine Differential Geometry [in Russian], Fizmatgiz, Moscow (1959).

    Google Scholar 

  24. B. Y. Chen. Geometry of Submanifolds, Marcel Dekker, New York (1973).

    Google Scholar 

  25. F. Fabricius-Bierre, “Sur varietes a torsion nulle,” Acta Math.,66, 49–77 (1936).

    Google Scholar 

  26. W. Klingenberg, “Zur affinen Differentialgeometrie. I. Über p-dimensionale Minimal-flächen und Sphären in n-dimensionalen Raum,” Math. Z.,54, No. 1, 65–80 (1951).

    Google Scholar 

Download references

Authors

Additional information

Translated from Itogi Nauki i Tekhniki, Seriya Problemy Geometrii, Vol. 21, pp.93–107, 1989.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chakmazyan, A.V. Normal connection in the geometry of normalized submanifolds of affine space. J Math Sci 55, 2131–2140 (1991). https://doi.org/10.1007/BF01095907

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01095907

Keywords

Navigation