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

Information Geometry of q-Gaussian Densities and Behaviors of Solutions to Related Diffusion Equations