Killing helices on a symmetric space of rank one (Q2490668)

The author studies helices on \(\mathbb K\)-space forms. A helix of proper order \(d\) on a Riemannian manifold \(M\) is a smooth curve \(\gamma \) for which there exists an orthonormal system of vector fields \((V_1=\dot \gamma ,V_2,\ldots ,V_d)\), and constants \(\kappa _i\), such that \[ \nabla _{\dot \gamma } V_i(t) =-\kappa _{i-1} V_{i-1}(t)+\kappa _iV_{i+1}(t) \] (\(1\leq i \leq d\), \(V_0=V_{d+1}=0\), \(\kappa _0=\kappa _d=0\)). For a helix \(\gamma \) on a Riemannian manifold with a \(\mathbb K\)-structure (\({\mathbb K}={\mathbb C,\mathbb H}\) or \(\mathbb O\)) structure torsion fields \(\widehat \tau _{ij}\) along \(\gamma \) are defined, and it is said that the helix \(\gamma \) is Killing if all its structure torsion fields are parallel. Two helices \(\gamma ,\sigma \) on a Riemannian manifold with a \(\mathbb K\)-structure are said to be \(\mathbb K\)-congruent if there is a \(\mathbb K\)-isometry \(\varphi \) and \(t_0\in {\mathbb R}\) such that \(\sigma (t)=\varphi \circ \gamma (t+t_0)\). For a \(\mathbb K\)-space form \(M\), i.e., a rank one Riemannian symmetric space with a \(\mathbb K\)-structure, the author determines under which conditions two helices are \(\mathbb K\)-congruent, and for \(d=1,2,3\), he determines the moduli space \({\mathcal M}_d(M)\), which is the set of \(\mathbb K\)-congruence classes of Killing helices of proper order \(d\).