  A theoretical analysis of active motion on curved surfaces is presented in
terms of a generalization of the Telegrapher's equation. Such generalized
equation is explicitly derived as the polar approximation of the hierarchy of
equations obtained from the corresponding Fokker-Planck equation of active
particles diffusing on curved surfaces. The general solution to the generalized
telegrapher's equation is given for a pulse with vanishing current as initial
data. Expressions for the probability density and the mean squared
geodesic-displacement are given in the limit of weak curvature. As an explicit
example of the formulated theory, the case of active motion on the sphere is
presented, where oscillations observed in the mean squared
geodesic-displacement are explained.
