Geodesic
Geometry of the Grassmann image of a local isometric immersion of Lobachevskii $n$-dimensional isometric immersion of Lobachevskii $n$-dimensional
Sbornik. Mathematics, Tome 188 (1997) no. 1, pp. 1-27
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In this paper we obtain an expression for the curvature tenser of the metric of the Grassmann image of an isometric immersion of Lobachevskii $n$-dimensional space in $(2n-1)$-dimensional Euclidean space. The main result is the following theorem: there is no $C^3$ local isometric immersion of 3-dimensional Lobachevski space in 5-dimensional Euclidean space with constant curvature of the metric of the Grassmann image. 
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Yu. A. Aminov. Geometry of the Grassmann image of a local isometric immersion of Lobachevskii $n$-dimensional isometric immersion of Lobachevskii $n$-dimensional. Sbornik. Mathematics, Tome 188 (1997) no. 1, pp. 1-27. http://geodesic.mathdoc.fr/item/SM_1997_188_1_a0/

[1] Moore J. D., “Isometric immeersions of space forms in space forms”, Pacific J. Math., 40 (1972), 157–166 | MR | Zbl

[2] Aminov Yu. A., “O pogruzhenii oblastei $n$-mernogo prostranstva Lobachevskogo v $(2n-\nobreak 1)$-mernoe evklidovo prostranstvo”, Dokl. AN SSSR, 236:3 (1977), 521–524 | MR | Zbl

[3] Aminov Yu. A., “Izometricheskie pogruzheniya oblastei $n$-mernogo prostranstva Lobachevskogo v $(2n-1)$-mernoe evklidovo prostranstvo”, Matem. sb., 111:3 (1980), 402–433 | MR | Zbl

[4] Tenenblat K., Terng C.-L., “A higher dimension generalization of the sine-Gordon equation and its Backlund transformation”, Bull. Amer. Math. Soc. (N.S.), 1 (1979), 589–593 | DOI | MR | Zbl

[5] Tenenblat K., Terng C.-L., “Backlund theorem for $n$-dimensional submanifolds of $\mathbb R^{2n-1}$”, Ann. of Math., 111 (1980), 477–490 | DOI | MR | Zbl

[6] Aminov Yu. A., “Izometricheskie pogruzheniya s ploskoi normalnoi svyaznostyu oblastei $n$-mernogo prostranstva Lobachevskogo v evklidovy prostranstva. Model kalibrovochnogo polya”, Matem. sb., 137 (179):3 (1988), 275–299 | MR | Zbl

[7] Aminov Yu. A., “Svoistva grassmanova obraza lokalnogo pogruzheniya oblasti trekhmernogo prostranstva Lobachevskogo v pyatimernoe evklidovo prostranstvo”, Ukr. geom. sb., 1984, no. 27, 3–11 | MR | Zbl

[8] Rabelo M. L., Tenenblat K., Toroidal submanifolds of constant non positive curvature, Preprint. Proceedings of conference N. I. Lobachevsky and Modern Geometry, Kazan, Russian | MR | Zbl

[9] Masaltsev L. A., “Psevdosfericheskie kongruentsii Bianki v $E^{2n-1}$”, Matem. fizika, analiz, geometriya, 1:3/4 (1994), 505–512 | MR | Zbl

[10] Aminov Y., Rabelo M. L., “On toroidal submanifolds of constant negative curvature”, Matem. fizika, analiz, geometriya, 2:3 (1995), 502–513 | MR

[11] Eizenkhart L. P., Rimanova geometriya, IL, M., 1948

[12] Egorov D. F., Raboty po differentsialnoi geometrii, Nauka, M., 1970 | MR