If we consider a new probability distribution defined through $f(x;\theta_i)$ and $g(x;\tilde\theta_i)$ given by $P(x;\theta_i,\tilde\theta_i)=\sqrt{f(x;\theta_i)g(x;\tilde\theta_i)}$, we can generalize the Fisher-Rao metric in the following way.
$$\frac{\partial H}{\partial\theta_i}=-\frac{1}{2}\int_\Omega dxP(x;\theta_i,\tilde\theta_i)\frac{\partial}{\partial\theta_i}\ln f(x;\theta_i)$$
$$\frac{\partial H}{\partial\tilde\theta_i}=-\frac{1}{2}\int_\Omega dxP(x;\theta_i,\tilde\theta_i)\frac{\partial}{\partial\tilde\theta_i}\ln g(x;\tilde\theta_i)$$
and so
$$\frac{\partial^2 H}{\partial\theta_i\partial\theta_j}=-\frac{1}{2}\int_\Omega dxP(x;\theta_i,\tilde\theta_i)\left[\frac{\partial^2}{\partial\theta_i\partial\theta_j}\ln f(x;\theta_i)+\frac{1}{4}\frac{\partial}{\partial\theta_i}\ln f(x;\theta_i)\frac{\partial}{\partial\theta_j}\ln f(x;\theta_i)\right]$$
and similarly
$$\frac{\partial^2 H}{\partial\tilde\theta_i\partial\tilde\theta_j}=-\frac{1}{2}\int_\Omega dxP(x;\theta_i,\tilde\theta_i)\left[\frac{\partial^2}{\partial\tilde\theta_i\partial\tilde\theta_j}\ln g(x;\tilde\theta_i)+\frac{1}{4}\frac{\partial}{\partial\tilde\theta_i}\ln g(x;\tilde\theta_i)\frac{\partial}{\partial\tilde\theta_j}\ln g(x;\tilde\theta_i)\right].$$
One has also cross products as
$$\frac{\partial^2 H}{\partial\theta_i\partial\tilde\theta_j}=-\frac{1}{4}\int_\Omega dxP(x;\theta_i,\tilde\theta_i)\frac{\partial}{\partial\theta_i}\ln f(x;\theta_i)\frac{\partial}{\partial\tilde\theta_j}\ln g(x;\tilde\theta_i).$$
Now, as done by Rao about Fisher information matrix, we can interpret these second derivatives of $H$ as the components of a metric tensor $h_{ij}(\theta,\tilde\theta)$ so that we can write down
$$ds^2=h_{ij}(\theta,\tilde\theta)d\theta_id\tilde\theta_j$$
and work out Riemann geometry.