I'm not very familiar with the Riemannian geometry. I want to know how to derive the geodesic equation for the following manifold of the positive orthant:
$$\mathbb{R}_{+}^{n}=\left\{x \in \mathbb{R}^{n}: x_{1}, \ldots, x_{n}>0\right\}$$
Consider a map $\varphi(x)=\log (x)=\left(\log \left(x_{1}\right), \ldots, \log \left(x_{n}\right)\right)$, then by the Frechet derivative, we can derive the differential:
$$d \varphi_x(u)=\left(\frac{u_{1}}{x_{1}}, \ldots, \frac{u_{n}}{x_{n}}\right).$$
We consider the following inner product:
$$\langle u, v\rangle_{x}^{+} \triangleq\langle\mathrm{D} \varphi(x)[u], \mathrm{D} \varphi(x)[v]\rangle=\sum_{i=1}^{n} \frac{u_{i} v_{i}}{x_{i}^{2}}$$
(I understand until this stage.) From the book, it says that the geodesic equation between $x,x'\in\mathbb R_+^n$ is given by
$$c(t)=\exp \left(\log (x)+t\left(\log \left(x^{\prime}\right)-\log (x)\right)\right)$$
where $t\in [0,1]$. Anyone to help derive the geodesic equation or any reference?