On the extrinsic geometric properties of parabolic surfaces and topological properties of saddle surfaces in symmetric spaces of rank one

A. A. Borisenko

Sbornik. Mathematics, Tome 44 (1983) no. 3, pp. 401-415

Résumé

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