Lengths of circular trajectories on geodesic spheres in a complex projective space

Behaviors of circular trajectories on hypersurfaces of type (A1) in a complex hyperbolic space

“…This cubic equation coincides with the characteristic equation for circles on CP n ð4Þ of geodesic curvature 1= ffiffi ffi 2 p and complex torsion t 12 ¼ t G ðk; rÞ (see (5.1) in [8]). By use of Proposition 4 in [8] (see also [4]) we get the following.…”

“…We need to consider the case that three conditions zðk; rÞ > 0, jt T ðk; rÞj a 1 and jkj > tanh r hold. We compare (6.3) with the characteristic equation for circles on CP n ð4Þ of geodesic curvature 1= ffiffi ffi 2 p and complex torsion t 12 ¼ t T ðk; rÞ (see (5.1) in [8]). If we consider the case t T ðk; rÞ ¼ 0 we obtain the first and the second assertions.…”

“…In the preceding paper [7], we showed there exist infinitely many circular trajectories which are not congruent to each other on geodesic spheres in a complex projective space CP n and horospheres, geodesic spheres and tubes around complex hypersurfaces congruent to CH nÀ1 in CH n . As we studied properties of circular trajectories on geodesic spheres in a complex projective space in [8], we here treat circular trajectories on other hypersurfaces in complex hyperbolic spaces. On horospheres and on tubes around complex hypersurfaces, because they are noncompact, we first study whether circular trajectories are bounded or not (Theorems 1 and 4).…”

Structure torsions of trajectories on real hypersurfaces of exceptional type in complex projective space

“…They characterize these real hypersurfaces by properties of circular trajectories. In their papers [3] and [4] they estimated the length of circular trajectories on real hypersurface of type (A 1 ) in a complex projective space and in a complex hyperbolic space. They studied the trajectories under Sasakian magnetic field on real hypersurfaces of type (B) in a complex hyperbolic space in [5] and [6].…”