COMPLETE $ l$-DIMENSIONAL SURFACES OF NONPOSITIVE EXTRINSIC CURVATURE IN A RIEMANNIAN SPACE

Abstract

This article studies complete -dimensional surfaces of nonpositive extrinsic 2-dimensional sectional curvature and nonpositive -dimensional curvature (for even) in Euclidean space , in the sphere , in the complex projective space , and in a Riemannian space . If the embedding codimension is sufficiently small, then a compact surface in or is a totally geodesic great sphere or complex projective space, respectively. If is a compact surface of negative extrinsic 2-dimensional curvature in a Riemannian space , then there are restrictions on the topological type of the surface. It is shown that a compact Riemannian manifold of nonpositive -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 -dimensional curvature in Euclidean space is estimated; it is determined when such surfaces are cylinders. A question about surfaces in which are homeomorphic to a sphere and which have nonpositive extrinsic curvature is looked at.Bibliography: 25 titles.