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Mat. Sb. (N.S.), 1970, Volume 83(125), Number 2(10), Pages 313–324 (Mi msb3514)  

This article is cited in 3 scientific papers (total in 3 papers)

Some properties of surfaces with slowly varying negative extrinsic curvature in a Riemannian space

I. S. Brandt


Abstract: 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:
$$ \sup_F\{|\operatorname{grad}\frac1k|+\frac{\Lambda-\lambda}{2k^2}\}=q>\frac1{\sqrt3}, $$
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.

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English version:
Mathematics of the USSR-Sbornik, 1970, 12:2, 313–324

Bibliographic databases:

UDC: 513.736.3
MSC: 14H55, 53C21, 14Jxx
Received: 16.03.1970

Citation: I. S. Brandt, “Some properties of surfaces with slowly varying negative extrinsic curvature in a Riemannian space”, Mat. Sb. (N.S.), 83(125):2(10) (1970), 313–324; Math. USSR-Sb., 12:2 (1970), 313–324

Citation in format AMSBIB
\Bibitem{Bra70}
\by I.~S.~Brandt
\paper Some properties of surfaces with slowly varying negative extrinsic curvature in a~Riemannian space
\jour Mat. Sb. (N.S.)
\yr 1970
\vol 83(125)
\issue 2(10)
\pages 313--324
\mathnet{http://mi.mathnet.ru/msb3514}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=268824}
\zmath{https://zbmath.org/?q=an:0203.24303}
\transl
\jour Math. USSR-Sb.
\yr 1970
\vol 12
\issue 2
\pages 313--324
\crossref{https://doi.org/10.1070/SM1970v012n02ABEH000923}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. I. V. Gribkov, “The problem of the correctness of Schur's theorem”, Math. USSR-Sb., 44:4 (1983), 471–481  mathnet  crossref  mathscinet  zmath
    2. I. V. Gribkov, “The multidimensional problem of the correctness of Schur's theorem”, Math. USSR-Sb., 48:2 (1984), 423–436  mathnet  crossref  mathscinet  zmath
    3. Connell Ch., Ullman J., “Ends of Negatively Curved Surfaces in Euclidean Space”, Manuscr. Math., 131:3-4 (2010), 275–303  crossref  mathscinet  zmath  isi
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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