Geodesic
Internal geometry of hypersurfaces in projectively metric space
Fundamentalʹnaâ i prikladnaâ matematika, Tome 16 (2010) no. 2, pp. 103-114.

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In this paper, we study the internal geometry of a hypersurface $\mathrm V_{n-1}$ embedded in a projectively metric space $\mathrm K_n$, $n\ge3$, and equipped with fields of geometric-objects $\{G^i_n,G_i\}$ and $\{H^i_n,G_i\}$ in the sense of Norden and with a field of a geometric object $\{H^i_n,H_n\}$ in the sense of Cartan. For example, we have proved that the projective-connection space $\mathrm P_{n-1, n-1}$ induced by the equipment of the hypersurface $\mathrm V_{n-1}\subset\mathrm K_n$, $n\ge3$, in the sense of Cartan with the field of a geometrical object $\{H^i_n,H_n\}$ is flat if and only if its normalization by the field of the object $\{H^i_n,G_i\}$ in the tangent bundle induces a Riemannian space $R_{n-1}$ of constant curvature $\mathrm K=-1/c$. 
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A. V. Stolyarov. Internal geometry of hypersurfaces in projectively metric space. Fundamentalʹnaâ i prikladnaâ matematika, Tome 16 (2010) no. 2, pp. 103-114. http://geodesic.mathdoc.fr/item/FPM_2010_16_2_a10/

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

[2] Evtushik L. E., Lumiste Yu. G., Ostianu N. M., Shirokov A. P., “Differentsialno-geometricheskie struktury na mnogoobraziyakh”, Itogi nauki i tekhn. Ser. Probl. geom. Tr. geom. sem., 9, 1979, 5–246 | MR | Zbl

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

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

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

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

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

[8] Stolyarov A. V., “Dvoistvennye proektivno-metricheskie prostranstva, opredelyaemye regulyarnoi giperpoverkhnostyu”, Vestn. Chuvash. gos. ped. un-ta im. I. Ya. Yakovleva, 2009, no. 1(61), 29–36 | MR

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

[10] Cartan E., “Les éspaces a connexion projective”, Trudy seminara po vektornomu i tenzornomu analizu MGU, 4, 1937, 147–159 | Zbl