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This article is cited in 5 scientific papers (total in 5 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:
$$
\sup_F\biggl\{\biggl|\operatorname{grad}\frac1k\biggr|+\frac{\Lambda-\lambda}{2k^2}\biggr\}=q>\frac1{\sqrt3},
$$
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.
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
Linking options:
https://www.mathnet.ru/eng/sm3514https://doi.org/10.1070/SM1970v012n02ABEH000923 https://www.mathnet.ru/eng/sm/v125/i2/p313
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Abstract page: | 204 | Russian version PDF: | 66 | English version PDF: | 2 | References: | 24 |
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