Dual Riemannian spaces of constant curvature on a~normalized hypersurface
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 11 (2010), pp. 63-73.

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

In this paper we obtain the following results: 1) we prove that in a differential neighborhood of the fourth order a regular hypersurface $\mathrm V_{n-1}$ embedded in a projective-metric space $\mathrm K_n$, $n\geqslant3$, intrinsically induces the dual projective-metric space $\overline K_n$; 2) we obtain an invariant analytical condition under which the normalization of the hypersurface $\mathrm V_{n-1}\subset\mathrm K_n$ (the tangential hypersurface $\overline{\mathrm V}_{n-1}\subset\overline{\mathrm K}_n$) by fields of quasitensors $H^i_n$, $H_i$ ($\overline H^i_n$, $\overline H_i$) induces a Riemannian space of constant curvature. Note that when these two conditions are fulfilled simultaneously, spaces $R_{n-1}$ and $\overline R_{n-1}$ are dual with the same identical constant curvature $\mathrm K=-\frac1c$; 3) we give geometric descriptions of the obtained analytical conditions.
Keywords: projective-metric space, duality, normalization, Riemannian connection, Riemannian space of constant curvature.
@article{IVM_2010_11_a5,
     author = {A. V. Stolyarov},
     title = {Dual {Riemannian} spaces of constant curvature on a~normalized hypersurface},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {63--73},
     publisher = {mathdoc},
     number = {11},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2010_11_a5/}
}
TY  - JOUR
AU  - A. V. Stolyarov
TI  - Dual Riemannian spaces of constant curvature on a~normalized hypersurface
JO  - Izvestiâ vysših učebnyh zavedenij. Matematika
PY  - 2010
SP  - 63
EP  - 73
IS  - 11
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IVM_2010_11_a5/
LA  - ru
ID  - IVM_2010_11_a5
ER  - 
%0 Journal Article
%A A. V. Stolyarov
%T Dual Riemannian spaces of constant curvature on a~normalized hypersurface
%J Izvestiâ vysših učebnyh zavedenij. Matematika
%D 2010
%P 63-73
%N 11
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IVM_2010_11_a5/
%G ru
%F IVM_2010_11_a5
A. V. Stolyarov. Dual Riemannian spaces of constant curvature on a~normalized hypersurface. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 11 (2010), pp. 63-73. http://geodesic.mathdoc.fr/item/IVM_2010_11_a5/

[1] Norden A. P., Prostranstva affinnoi svyaznosti, Nauka, M., 1976 | MR | Zbl

[2] Laptev G. F., “Differentsialnaya geometriya pogruzhennykh mnogoobrazii”, Tr. Mosk. matem. o-va, 2, 1953, 275–382 | MR | Zbl

[3] Finikov S. P., Metod vneshnikh form Kartana, GITTL, M.–L., 1948

[4] Stolyarov A. V., “Vnutrennyaya geometriya proektivno-metricheskogo prostranstva”, Differents. geometriya mnogoobrazii figur, 32, Kaliningrad, 2001, 94–101 | MR | Zbl

[5] Stolyarov A. V., Dvoistvennaya teoriya osnaschennykh mnogoobrazii, Chuvash. gos. ped. un-t im. I. Ya. Yakovleva, Cheboksary, 1994 | MR | Zbl

[6] Abrukov D. A., Vnutrennyaya geometriya poverkhnostei i raspredelenii v proektivno-metricheskom prostranstve, Chuvash. gos. ped. un-t im. I. Ya. Yakovleva, Cheboksary, 2003

[7] Evtushik L. E., Lumiste Yu. G., Ostianu N. M., Shirokov A. P., Differentsialno-geometricheskie struktury na mnogoobraziyakh, Itogi nauki i tekhniki. Problemy geometrii, 9, VINITI, M., 1979 | MR | Zbl

[8] Rashevskii P. K., Rimanova geometriya i tenzornyi analiz, Nauka, M., 1967 | MR

[9] Kobayasi Sh., Nomidzu K., Osnovy differentsialnoi geometrii, v. 1, Nauka, M., 1981