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.
Similar content being viewed by others
Literature cited
M. A. Akivis, “Focal forms of a surface of rank r,” Izv. Vyssh. Uchebn. Zaved., Mat., No. 1, 9–19 (1957).
M. A. Akivis, “Multidimensional surfaces supporting a net of conjugate lines,” Dokl. Akad. Nauk SSSR,139, No. 6, 1279–1282 (1961).
M. A. Akivis, “Voss normals of a surface supporting a net of conjugate lines,” Mat. Shorn.,58, No. 2, 695–706 (1962).
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).
E. D. Alshibaya, “Differential geometry of a hypersurface in a multidimensional affine space,” Tr. Tbilis. Univ.,129, 319–341 (1968).
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).
V. T. Bazylev, “Multidimensional nets and their transformations,” Itogi Nauki Tekh., Ser. Geom., 1963, 138–164 (1965).
E. Cartan, Riemannian Geometry in an Orthonormal Frame [Russian translation], MGU, Moscow (1960).
G. F. Laptev, “Invariant normalization of a surface in a space with affine connection,” Dokl. Akad. Nauk SSSR,126, No. 3, 490–493 (1959).
Yu. G. Lumiste, “Differential geometry of submanifolds,” Itogi Nauki Tekh., Ser, Alg. Topol. Geom.,13, 273–340 (1975).
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).
K. Nomidzu, Lie Group and Differential Geometry [Russian translation], Moscow (1960).
A. V. Norden, Spaces with Affine Connection [in Russian], 2nd edition, Nauka, Moscow (1976).
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).
D. I. Perepelkin, “Curvature and normal spaces of a manifold Vm in Rn,” Mat. Sborn.,42, No. 1, 81–120 (1935).
D. I. Perepelkin, “Parallel submanifolds in Euclidean (or Riemannian) spaces,” Dokl. Akad. Nauk SSSR,1, 593–598 (1935).
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.
A. V. Chakmazyan, “Normalized submanifolds of affine space with flat normal affine connection,” in: Differential Geometry [in Russian], Kalinin (1977), pp. 120–129.
A, V. Chakmazyan, “Connection in normal bundles of a normal submanifold Vm in Pn,” Itogi Nauki Tekh. Ser. Probl. Geom.,10, 55–75 (1978).
A. V. Chakmazyan, “Normalized submanifolds with flat normal connection in projective space,” Mat. Zametki,33, No. 2, 281–288 (1983).
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).
A. V. Chakmazyan, “Normalizations with flat normal connection for submanifolds of affine space,” Izv. Vuzov. Matematika,1, 74–79 (1987).
P. A. Shirokov and A. P. Shirokov, Affine Differential Geometry [in Russian], Fizmatgiz, Moscow (1959).
B. Y. Chen. Geometry of Submanifolds, Marcel Dekker, New York (1973).
F. Fabricius-Bierre, “Sur varietes a torsion nulle,” Acta Math.,66, 49–77 (1936).
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).
Additional information
Translated from Itogi Nauki i Tekhniki, Seriya Problemy Geometrii, Vol. 21, pp.93–107, 1989.
Rights 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
Issue Date:
DOI: https://doi.org/10.1007/BF01095907