  In this paper we present a geometric interpretation of the periodic
Degasperis-Procesi equation as the geodesic flow of a right invariant symmetric
linear connection on the diffeomorphism group of the circle. We also show that
for any evolution in the family of $b$-equations there is neither gain nor loss
of the spatial regularity of solutions. This in turn allows us to view the
Degasperis-Procesi and the Camassa-Holm equation as an ODE on the Fr\'echet
space of all smooth functions on the circle.
