I am stuck by the following problem.
Let $ g = 1/y^2 (dx \otimes dx + dy \otimes dy)$ be a Riemannian metric on the half-plane $ S = \mathbb{R}^2_{y > 0}$. What is the explicit form of the exponential map $exp_p: T_p(S) \to S$ when $p$ is the point $(0,1)$?
Also, write down the inverse of $exp_p$, the geodesic normal coordinates, and the pullback $(exp_p)^* g $ of the metric.
I know the exponential map $exp_p(v) = \gamma_v(1)$, where $\gamma_v$ is the maximal geodesic with velocity $v$ at starting at $p$. In $S$, the geodesics are the vertical lines and the half-circles centered at the $x$-axis.
But how do I write down explicitly the exponential map?
Edit: I found the equation for the geodesics to be $L(t)=(a,be^t)$ (for the vertical lines) and $C(t) = (a \tanh t + b, \frac{a}{\cosh t})$ (for circles centered on the $x$-axis). For $L(t)$, to start at $(0,1)$ it must be $(0,e^t)$, from which its initial velocity is $0$. But I don't know how to rewrite $C(t)$ in such a way that it starts at $(0,1)$.