Suppose, $D$ is the space of all continuous distributions over $\mathbb{R}$, equipped with Jensen-Shannon metric:
$$d(P, Q) = \sqrt{\int_{-\infty}^\infty (p(t)\ln(1 + \frac{p(t)-q(t)}{p(t)+q(t)}) + q(t)\ln(1 - \frac{p(t)-q(t)}{p(t)+q(t)}))dt}$$
Here $p$ and $q$ are PDFs of $P$ and $Q$.
Now, suppose $P(\theta): [0;1] \to D$ is a smooth curve in $D$, and $p(t, \theta)$ is a parametric PDF corresponding to it. Then for $\epsilon \to 0$ we have $d(P(\theta), P(\theta+\epsilon)) = \sqrt{\int_{-\infty}^\infty (p(\ln(1 + \epsilon\frac{\partial p}{\partial \theta}\frac{1}{2p}) + \ln(1 - \epsilon\frac{\partial p}{\partial \theta}\frac{1}{2p})) + \epsilon \frac{\partial p}{\partial \theta} \ln(1 - \epsilon\frac{\partial p}{\partial \theta}\frac{1}{2p}) )dt} + O(\epsilon^2) = \epsilon\sqrt{\int_{-\infty}^\infty (-p(\frac{\partial p}{\partial \theta}\frac{1}{2p})^2 + \frac{(\frac{\partial p}{\partial \theta})^2}{2p}) )dt} + O(\epsilon^2) = \epsilon\sqrt{\int_{-\infty}^\infty \frac{(\frac{\partial p}{\partial \theta})^2}{4p} dt} + O(\epsilon^2)$. That means, that the length of the curve will be
$$\int_0^1\sqrt{\int_{-\infty}^\infty \frac{(\frac{\partial p}{\partial \theta})^2}{4p} dt}d\theta$$
My question is, what will be the form of geodesic curves in this metric space?
It seems, that describing them is equivalent to solving a minimization problem for
$$\int_0^1\sqrt{\int_{-\infty}^\infty \frac{(\frac{\partial p}{\partial \theta})^2}{4p} dt}d\theta$$
under constraints
$$p \geq 0$$
and
$$\int_{-\infty}^\infty pdt = 1$$
However, I have no idea how to solve it.