  This paper presents new geometric aspects of the behaviors of solutions to
the porous medium equation (PME) and its associated equation. First we discuss
the Legendre structure with information geometry on the manifold of generalized
exponential densities. Next by considering such a structure in particular on
the q-Gaussian densities, we derive several physically and geometrically
interesting properties of the solutions. They include, for example,
characterization of the moment-conserving projection of a solution, evaluation
of evolutional velocities of the second moments and the convergence rate to the
manifold in terms of the geodesic curves, divergence and so on.
