 Exponential displacement map on a compact surface

If $S$ is compact and $V$ is a smooth vector field, thenis smooth and well-defined by the Hopf-Rinow theorem. The maps ${\varphi }_t(p)={\exp }_p(tV(p))$ form a smooth homotopy from the identity to $\varphi$, so ${\mathrm{deg}}_2(\varphi )=1$.