Following is a question which I also asked in stats.stackexchange where I haven't got a response yet. Here is a link.
Let $S$ be the set of all probability distributions on $\mathbb{R}$ and $S_n=\{p_\theta\}$ be an $n$ dimensional submanifold of parameterized family of probability distributions on $\mathbb{R}$ where $\theta=(\theta_1,\dots, \theta_n)\in \Theta$ are called parameters of the probability density $p_\theta(x)$ and $\Theta$ is an open set homeomorphic to $\mathbb{R}^n$. It is also assumed that $p_\theta(x)>0$ for all $x$.
For example, $$p_{\theta=(\mu,\sigma)}(x)=\frac{1}{\sqrt{2\pi}\sigma}e^{\frac{-(x-\mu)^2}{2\sigma^2}}$$ where $\mu\in \mathbb{R},\sigma>0$ forms a $2$-dimensional submanifold.
On this manifold, a Riemannian metric is defined by the following, called Fisher information metric.
$$ g_{i,j}(\partial_i,\partial_j)=\int \partial_i \log p_\theta(x) ~\partial_j \log p_\theta(x) ~p_\theta(x)~dx$$ where $\partial_i=\frac{\partial}{\partial \theta_i}$.
My question is regarding a point in page 384 from this Ann. Stat. paper by Amari.
In the above mentioned paper, when he says a curve $\{q_t(x)\}$ in $S$, where $q_0(x)=p_\theta(x)$ intersects orthogonally $S_n$ at $p_\theta(x)$, he means $$\int \partial_i \log p_\theta(x) ~\frac{d}{dt} \log q_t(x)\lvert_{t=0} ~q_0(x)~dx=0$$.
Why? Could somebody make me understand by giving intuitive explanation of these concepts?