Geodesic
4-webs on hypersurfaces of 4-axial space
Fundamentalʹnaâ i prikladnaâ matematika, Tome 16 (2010) no. 1, pp. 65-79 
V. V. Zabrodin. 4-webs on hypersurfaces of 4-axial space. Fundamentalʹnaâ i prikladnaâ matematika, Tome 16 (2010) no. 1, pp. 65-79. http://geodesic.mathdoc.fr/item/FPM_2010_16_1_a5/
@article{FPM_2010_16_1_a5,
     author = {V. V. Zabrodin},
     title = {4-webs on hypersurfaces of 4-axial space},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {65--79},
     year = {2010},
     volume = {16},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2010_16_1_a5/}
}
TY  - JOUR
AU  - V. V. Zabrodin
TI  - 4-webs on hypersurfaces of 4-axial space
JO  - Fundamentalʹnaâ i prikladnaâ matematika
PY  - 2010
SP  - 65
EP  - 79
VL  - 16
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/FPM_2010_16_1_a5/
LA  - ru
ID  - FPM_2010_16_1_a5
ER  - 
%0 Journal Article
%A V. V. Zabrodin
%T 4-webs on hypersurfaces of 4-axial space
%J Fundamentalʹnaâ i prikladnaâ matematika
%D 2010
%P 65-79
%V 16
%N 1
%U http://geodesic.mathdoc.fr/item/FPM_2010_16_1_a5/
%G ru
%F FPM_2010_16_1_a5

Voir la notice de l'article provenant de la source Math-Net.Ru

V. B. Lazareva investigated 3-webs formed by shadow lines on a surface embedded in 3-dimensional projective space is assuming that the lighting sources are situated on 3 straight lines. The results were used, in particular, for the solution of Blaschke problem of classification of regular 3-webs formed by pencils of circles in a plane. In the present paper, we consider a 4-web $W$ formed by shadow surfaces on a hypersurface $V$ embedded in 4-dimensional projective space assuming that the lighting sources are situated on 4 straight lines. We call the projective 4-space with 4 fixed straight lines a 4-axial space. Structure equations of 4-axial space and of the surface $V$, asymptotic tensor of $V$, torsions and curvatures of 4-web $W$, and connection form of invariant affine connection associated with 4-web $W$ are found.

[1] Akivis M. A., “Lokalnye differentsiruemye kvazigruppy i tri-tkani mnogomernykh poverkhnostei”, Issledovaniya po teorii kvazigrupp i lup, Kishinëv, 1973, 3–12 | MR

[2] Akivis M. A., Mnogomernaya differentsialnaya geometriya, Kalinin, 1977 | MR | Zbl

[3] Akivis M. A., “O mnogomernom obobschenii seti Kënigsa”, Izv. vyssh. uchebn. zaved. Matematika, 1981, no. 9, 3–4 | MR | Zbl

[4] Blyashke V., Vvedenie v geometriyu tkanei, M., 1959

[5] Zabrodin V. V., “Strukturnye uravneniya 4-aksialnogo prostranstva” (to appear)

[6] Lazareva V. B., Tri-tkani na poverkhnostyakh v triaksialnom prostranstve, Kalinin, 1984

[7] Laptev G. F., “Differentsialnaya geometriya pogruzhënnykh mnogoobrazii”, Tr. MMO, 2, 1953, 275–382 | MR | Zbl

[8] Norden A. P., Prostranstva affinnoi svyaznosti, M., 1976 | MR

[9] Ryzhkov V. V., “Differentsialnaya geometriya tochechnykh sootvetstvii mezhdu prostranstvami”, Itogi nauki. Geometriya. 1963, 1965, 65–107 | MR

[10] Finikov S. P., Metod vneshnikh form Kartana v differentsialnoi geometrii, GITTL, M.–L., 1948

[11] Goldberg V. V., Theory of Multicodimensional $(n+1)$-Webs, Kluwer Academic, 1988 | MR | Zbl