Geometry of ordinary helices in a complex projective space

Bounded extrinsic curvature of subsets of metric spaces

“…Properties: (i) If X Riemannian manifold and Z ⊂ X a closed embedded submanifold then |II Z →X | ≤ λ if and only if Z is ( λ 2 24 , ρ)-convex for some sufficiently small ρ > 0 (ii) If X Riemannian manifold, the condition is equivalent to positive reach [15, Theorem 1.3] Subsets of R n+1 with uniformly positive reach can include, for example, finite disjoint unions of sets which are closed manifolds or manifolds with C 1 boundary of various dimensions which are joined along codimension ≥ 1 subsets of their boundaries. A (C, ρ)-convex subset need not have positive reach nor admit a supporting ball at each point, for a general ambient space.…”

“…For further discussion of (C, ρ)-convexity in relation to curves (especially in CBA spaces), see [3]. 2 Concerning spaces for which one can define λ-concave functions (such as Alexandrov spaces), one has Theorem 1. [5].…”

“…Circles of submanifolds of space forms (i.e. geodesics in the submanifold which are also round geometric circles in the ambient manifold) have been studied in [2], [1], etc. However, these works assumed the subset was a smooth (C 2 ) submanifold to begin with.…”

“…Proof. The equality analogue of property (i) is that |II Z →X | = λ if and only if Z is constantly (C = λ 2 24 , ρ)-convex. When n = 1, the first statement of the Proposition is true, since it is known that curves in R 2 of constant nonvanishing geodesic curvature are portions of standard round circles.…”

Killing helices on a symmetric space of rank one

“…For a Killing helix γ on CH 2 (−4), we consider its horizontal liftγ through Hopf fibration H 5 1 → CH 2 (−4). Regardingγ as a curve on C 3 , we find it satisfies the ordinary differential equatioñ γ (4) …”