Geodesic
On Isometric Immersions with Flat Normal Connection of the Hyperbolic Space~$L^n$ Into Euclidean Space~$E^{n+m}$
Matematičeskie zametki, Tome 82 (2007) no. 1, pp. 11-13.

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We prove that the hyperbolic space $L^n$ cannot be immersed in an Euclidean space $E^{n+m}$ with a flat normal connection provided the module of the mean curvature vector is bounded.
Keywords: immersion, mean curvature, flat normal connection, hyperbolic space, Grassmanian image, quasiisometric space.
Mots-clés : principal directions 
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D. V. Bolotov. On Isometric Immersions with Flat Normal Connection of the Hyperbolic Space~$L^n$ Into Euclidean Space~$E^{n+m}$. Matematičeskie zametki, Tome 82 (2007) no. 1, pp. 11-13. http://geodesic.mathdoc.fr/item/MZM_2007_82_1_a1/

[1] D. Hilbert, “Über Flächen von konstanter Gausscher Krummung”, Trans. Amer. Math. Soc., 2:1 (1901), 87–99 | DOI | MR | Zbl

[2] T. K. Milnor, “Efimov's theorem about complate immersed surfaces of negative curvature”, Adv. Math., 8:3 (1972), 474–543 | DOI | MR | Zbl

[3] N. V. Efimov, “Nepogruzhaemost poluploskosti Lobachevskogo”, Vestn. Mosk. un-ta. Ser. 1. Matem., mekh., 30:2 (1975), 83–86 | MR | Zbl

[4] S. Chern, N. H. Kuiper, “Some theorems on isometric imbedding of compact Riemann manifold in Euclidean space”, Ann. of Math. (2), 56 (1952), 422–430 | DOI | MR | Zbl

[5] Yu. A. Aminov, “Izometricheskie pogruzheniya s ploskoi normalnoi svyaznostyu oblastei $n$-mernogo prostranstva Lobachevskogo v evklidovo prostranstvo. Modeli teorii polya”, Matem. sb., 137:3 (1988), 275–299 | MR | Zbl

[6] Yu. A. Aminov, “Problemy vlozheniya: geometricheskie i topologicheskie aspekty”, Itogi nauki i tekhniki. Problemy geometrii, 13, VINITI, M., 1982, 119–156 | MR | Zbl

[7] Yu. A. Aminov, “Izometricheskie pogruzheniya oblasti v $3$-mernom prostranstve Lobachevskogo $L^3$ v evklidovo prostranstvo $E^5$ i dvizhenie tverdogo tela”, Matem. sb., 122:1 (1983), 12–30 | MR | Zbl

[8] Y. A. Nikolajevky, “A non-immersion theorem for a class of hyperbolic manifolds”, Differential Geom. Appl, 9:3 (1998), 239–242 | DOI | MR | Zbl

[9] F. Xavier, “A non-immersion theorem for hyperbolic manifolds”, Comment. Math. Helv., 60 (1985), 280–283 | DOI | MR | Zbl

[10] L. A. Masaltsev, “O minimalnykh podmnogoobraziyakh postoyannoi krivizny v evklidovom prostranstve”, Izv. vuzov. Ser. matem., 1998, no. 9, 64–65 | MR | Zbl

[11] D. Yu. Burago, Yu. D. Burago, S. V. Ivanov, Kurs metricheskoi geometrii, In-t kompyuter. issled., M.-Izhevsk, 2004