The problem is an abstract from applied science.

Given an $n$ dimensional Riemann manifold with metric $\langle M, g\rangle$, we could define deformation of the metric $g(t)$ where $t\in [0,1]$, for example, Ricci flow is such an deformation. And thus we should be able to measure the deforming energy(as an analog with kinetic energy in physics) point by point, and the total defroming energy may be in the form as

$E(t)=\int_{M} \left|\dfrac{\mathrm{d}R(t)}{\mathrm{d}t}\right|^n dA$

where $R$ indicate the Ricci curvature and $\left| \cdot \right|$ is a "proper" norm, such that the energy enjoys scale-invariant property. I will appreciate if someone could provide me some references.

Furthermore, if above formulation of energy is possible, we may define distance between two manifold $\langle M_1, g_1\rangle$ and $\langle M_2, g_2\rangle$ by

$d(M_1,M_2)=\mathrm{inf}_{g(t)}\int_0^1E(t)\mathrm{d} t$

where $g(0)=g_1$ and $g(1)=g_2$.