We consider a unit speed timelike curve $\alpha$ in Minkowski 4-space ${𝐄}_1^4$ and denote the Frenet frame of $\alpha$ by $\{𝐓,𝐍,{𝐁}_1,{𝐁}_2\}$. We say that $\alpha$ is a generalized helix if one of the unit vector fields of the Frenet frame has constant scalar product with a fixed direction $U$ of ${𝐄}_1^4$. In this work we study those helices where the function $\langle {𝐁}_2,U\rangle$ is constant and we give different characterizations of such curves.

We investigate totally umbilical submanifolds in manifolds satisfying some curvature conditions of either recurrent or pseudosymmetry type in the sense of Ryszard Deszcz and derive the respective condition for submanifolds. We also prove some relations involving the mean curvature and the Weyl conformal curvature tensor of submanifolds. Some examples are discussed.

The connected components of the zero set of any conformal vector field v, in a pseudo-Riemannian manifold (M, g) of arbitrary signature, are of two types, which may be called ‘essential’ and ‘nonessential’. The former consist of points at which v is essential, that is, cannot be turned into a Killing field by a local conformal change of the metric. In a component of the latter type, points at which v is nonessential form a relatively-open dense subset that is at the same time a totally umbilical...