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Mathematics of the USSR-Sbornik, 1977, Volume 33, Issue 4, Pages 485–499
DOI: https://doi.org/10.1070/SM1977v033n04ABEH002436
(Mi sm2979)
 

This article is cited in 15 scientific papers (total in 15 papers)

Complete $l$-dimensional surfaces of nonpositive extrinsic curvature in a Riemannian space

A. A. Borisenko
References:
Abstract: This article studies complete $l$-dimensional surfaces of nonpositive extrinsic 2-dimensional sectional curvature and nonpositive $k$-dimensional curvature (for $k$ even) in Euclidean space $E^n$, in the sphere $S^n$, in the complex projective space $\mathbf CP^n$, and in a Riemannian space $R^n$. If the embedding codimension is sufficiently small, then a compact surface in $S^n$ or $\mathbf CP^n$ is a totally geodesic great sphere or complex projective space, respectively. If $F^l$ is a compact surface of negative extrinsic 2-dimensional curvature in a Riemannian space $R^{2l-1}$, then there are restrictions on the topological type of the surface. It is shown that a compact Riemannian manifold of nonpositive $k$-dimensional curvature cannot be isometrically immersed as a surface of small codimension. The order of growth of the volume of complete noncompact surfaces of nonpositive $k$-dimensional curvature in Euclidean space is estimated; it is determined when such surfaces are cylinders. A question about surfaces in $S^3$ which are homeomorphic to a sphere and which have nonpositive extrinsic curvature is looked at.
Bibliography: 25 titles.
Received: 14.06.1977
Bibliographic databases:
UDC: 513.7
MSC: Primary 53C40; Secondary 53B25, 53A05
Language: English
Original paper language: Russian
Citation: A. A. Borisenko, “Complete $l$-dimensional surfaces of nonpositive extrinsic curvature in a Riemannian space”, Math. USSR-Sb., 33:4 (1977), 485–499
Citation in format AMSBIB
\Bibitem{Bor77}
\by A.~A.~Borisenko
\paper Complete $l$-dimensional surfaces of nonpositive extrinsic curvature in a~Riemannian space
\jour Math. USSR-Sb.
\yr 1977
\vol 33
\issue 4
\pages 485--499
\mathnet{http://mi.mathnet.ru//eng/sm2979}
\crossref{https://doi.org/10.1070/SM1977v033n04ABEH002436}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=470905}
\zmath{https://zbmath.org/?q=an:0372.53029|0397.53040}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=A1977GX43200003}
Linking options:
  • https://www.mathnet.ru/eng/sm2979
  • https://doi.org/10.1070/SM1977v033n04ABEH002436
  • https://www.mathnet.ru/eng/sm/v146/i4/p559
  • This publication is cited in the following 15 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics
    Statistics & downloads:
    Abstract page:409
    Russian version PDF:118
    English version PDF:29
    References:46
    First page:1
     
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