Extrinsic circular trajectories on totally η-umbilic real hypersurfaces in a complex hyperbolic space

Abstract

A trajectory for a Sasakian magnetic field, which is a generalization of geodesics, on a real hypersurface in a complex hyperbolic space CHⁿ is said to be extrinsic circular if it can be regarded as a circle as a curve in CHⁿ. We study how the moduli space of extrinsic circular trajectories, which is the set of their congruence classes, on a totally η-umbilic real hypersurface is contained in the moduli space of circles in CHⁿ. From this aspect we characterize tubes around totally geodesic complex hypersurfaces CH^n-1 in CHⁿ by some properties of such trajectories.