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Complete $l$-dimensional surfaces of nonpositive extrinsic curvature in a Riemannian space
A. A. Borisenko
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.
UDC:
513.7
MSC: Primary
53C40; Secondary
53B25,
53A05 Received: 14.06.1977
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