Torqued and Anti-Torqued Vector Fields on Hyperbolic Spaces

In this paper, we study the existence of torqued and anti-torqued vector fields on the hyperbolic ambient space $\mathbb{H}^n$. Although there are examples of proper torqued vector fields on open subsets on $\mathbb{H}^n$, we prove that there is no a proper torqued vector field globally defined on $\mathbb{H}^n$. Similarly, we have another non-existence result for anti-torqued vector fields as long as their conformal scalar is a non-constant function. When the conformal scalar is constant some examples of anti-torqued vector fields are provided.