Suppose we are on the two dimensional sphere $S^2$ the Riemannian metric $$(g_{ij})_{i,j}=\begin{bmatrix}1 & 0 \\ 0 & \sin^2(\theta)\end{bmatrix}$$ in spherical local coordinates ($\theta \in (0,\pi)$, $\varphi\in (0,2\pi)$). Is it possible to have an explicit local expression in spherical coordinates of the exponential map?
This is my attempt: using this formula given in $\mathbb{R}^3$
For a point $p\in S^2\subset \mathbb{R}^3$ and a direction $v\in \mathbb{R}^3$, the exponential map is given by \begin{equation}\exp_p(v)=\cos(|v|)p+\sin(|v|)\frac{v}{|v|},\end{equation} where $|\cdot|$ is the usual norm in $\mathbb{R}^3$.
I can convert $p=(p^1,p^2)$ and $v=v^1\partial_\theta+v^2\partial_\phi$ in local spherical coordinates, obtaining
\begin{equation}\exp_p(v)=\Bigl(p^1\cos(|v|)+\frac{v^1}{|v|},p^2\sin(|v|)+\frac{v^2}{|v|}\Bigr),\end{equation} where $|\cdot|$ is the norm induced by the Riemannian metric.
Is this correct?