Geodesic
Some properties of surfaces with slowly varying negative extrinsic curvature in a Riemannian space
Sbornik. Mathematics, Tome 12 (1970) no. 2, pp. 313-324 
I. S. Brandt. Some properties of surfaces with slowly varying negative extrinsic curvature in a Riemannian space. Sbornik. Mathematics, Tome 12 (1970) no. 2, pp. 313-324. http://geodesic.mathdoc.fr/item/SM_1970_12_2_a9/
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     author = {I. S. Brandt},
     title = {Some properties of surfaces with slowly varying negative extrinsic curvature in {a~Riemannian} space},
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Voir la notice de l'article provenant de la source Math-Net.Ru

We consider surfaces of negative extrinsic curvature in a Riemannian space with nonpositive curvature. We prove that the following inequality holds on a surface which is complete in the sense of the intrinsic metric: $$ \sup_F\biggl\{\biggl|\operatorname{grad}\frac1k\biggr|+\frac{\Lambda-\lambda}{2k^2}\biggr\}=q>\frac1{\sqrt3}, $$ here $F$ is the surface being considered, $k=\sqrt{K_e}$ ($K_e$ is the extrinsic curvature of $F$) and $\Lambda$ and $\lambda$ are the maximum and minimum of the Riemannian curvature of the space $R$ at a given point. This theorem generalizes a theorem of Efimov concerning $q$-metrics. We give an example of a surface for which $q=4,5$. Bibliography: 8 titles.

[1] N. V. Efimov, “Poverkhnosti s medlenno izmenyayuscheisya otritsatelnoi kriviznoi”, Uspekhi matem. nauk, XXI:5(131) (1966), 3–58 | MR

[2] N. V. Efimov, “Differentsialnye priznaki gomeomorfnosti nekotorykh otobrazhenii s primeneniem v teorii poverkhnostei”, Matem. sb., 76(118) (1968), 499–512 | MR | Zbl

[3] N. V. Efimov, E. G. Poznyak, “Nekotorye preobrazovaniya osnovnykh uravnenii teorii poverkhnostei”, DAN SSSR, 137:1 (1961), 25–27 | MR

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

[5] E. R. Rozendorn, “Svoistva asimptoticheskikh linii na poverkhnosti s medlenno menyayuscheisya otritsatelnoi kriviznoi”, DAN SSSR, 145:3 (1962), 538–540 | MR | Zbl

[6] L. P. Eizenkhart, Rimanova geometriya, IL, Moskva, 1948

[7] A. V. Pogorelov, Vneshnyaya geometriya vypuklykh poverkhnostei, Nauka, Moskva, 1969 | MR

[8] E. Kartan, Geometriya rimanovykh prostranstv, ONTI, Moskva–Leningrad, 1936