On proper helices and extrinsic spheres in pseudo-Riemannian geometry (Q1375791)

Let \(M\) be a pseudo-Riemannian manifold and \(C\) be a regular curve in \(M\). Then \(C\) is called a proper helix of order \(d\), if it has constant curvatures of osculating order \(d\) with respect to a certain Frénet frame. The authors investigate proper helices of order \(d\) in totally umbilical pseudo-Riemannian submanifolds. We mention here one of the main results. Let \(N\) be a totally umbilical pseudo-Riemannian submanifold of \(M\) such that each proper helix of order \(d\) in \(N\) is a proper helix of order \(d\) in \(M\). Then we have: (i) if \(d\) is odd, then \(M\) is totally geodesic; (ii) if \(d\) is even, then \(M\) is an extrinsic sphere. Special attention is given to the study of proper helices in extrinsic spheres.