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

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.
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