Torqued and Anti-Torqued Vector Fields on Hyperbolic Spaces

Abstract

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 of ${\mathbb{H}}^n$, we prove that there is not 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 satisfying certain conditions. When the conformal scalar is constant, some examples of anti-torqued vector fields are provided.

11 pages