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

A. A. Borisenko

Sbornik. Mathematics, Tome 33 (1977) no. 4, pp. 485-499

Résumé

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.