  We propose a geometric perspective to describe the motion of self-propelled
particles moving at constant speed in d dimensions. We exploit the fact that
the vector that conveys the direction of motion of the particle performs a
random walk on a $(d-1)$-dimensional manifold. We show that the particle
performs isotropic diffusion in d-dimensions if the manifold corresponds to a
hypersphere. In contrast, we find that the self-propelled particle exhibits
anisotropic diffusion if this manifold corresponds to a deformed hypersphere
(e.g. an ellipsoid). This simple approach provides an unified framework to deal
with isotropic as well as anisotropic diffusion of particles moving at constant
speed in any dimension.
