ON THE EXTRINSIC GEOMETRIC PROPERTIES OF PARABOLIC SURFACES AND TOPOLOGICAL PROPERTIES OF SADDLE SURFACES IN SYMMETRIC SPACES OF RANK ONE

Abstract

This paper investigates the metric structure of compact -parabolic surfaces and topological properties of -saddle surfaces in the sense of Sefel' in symmetric spaces of rank one, namely, spherical space , complex projective space , and quaternion projective space . It turns out that -parabolic surfaces for large are totally geodesic spheres in , totally geodesic complex projective spaces in , and totally geodesic quaternion projective spaces in . It follows that surfaces of nonpositive extrinsic -dimensional curvature, under a natural restriction on the codimension of the embedding, are totally geodesic surfaces in , and . Saddle surfaces for small have restrictions on the homology and cohomology groups. Since surfaces of nonpositive -dimensional extrinsic curvature for small codimension of the embedding are -saddle surfaces, they also have degeneracies in the homology and cohomology groups.Bibliography: 27 titles.