Prove that $\exp_0:\mathbb{C}\to \mathbb{D}$ is not holomorphic, where $\mathbb{D}$ is the unit disk with hyperbolic metric $ds^2=\frac{4dzd\bar{z}}{(1-|z|^2)^2}$.
I am reading Jost's Compact Riemann Surfaces. The exponential map is defined by the geodesic, to be more specific $$ \exp_p:v\mapsto \gamma_{p,v}(1), $$ where $\|v\|_p$ is suffiently small so the geodisic can be defined in $[0,1]$. Jost claims the exponential map is generally not holomorphic. But I do not know how to prove the problem. Can we just compute the expression of $\exp_0$? Appreciate any help or hint! This part is in p32-33 of the book.
Hint: For your concrete question, the canonical coordinate on $\mathbb C$ gives you a complex coordinate on $\mathbb D$ by restriction (recall that the complex coordinates on Riemannian surfaces come from the isothermal coordinates), and then just apply the Liouville's theorem to $\exp$.
For general cases, you can refer to this.