  Group equivariant non-expansive operators have been recently proposed as
basic components in topological data analysis and deep learning. In this paper
we study some geometric properties of the spaces of group equivariant operators
and show how a space $\mathcal{F}$ of group equivariant non-expansive operators
can be endowed with the structure of a Riemannian manifold, so making available
the use of gradient descent methods for the minimization of cost functions on
$\mathcal{F}$. As an application of this approach, we also describe a procedure
to select a finite set of representative group equivariant non-expansive
operators in the considered manifold.
