 Solution

The induced metric makes the compact surface $S$ a compact Riemannian manifold. By the Hopf-Rinow theorem it is geodesically complete, so every ${\exp }_p$ is defined on all of $T_{p}S$. In particular, ${\exp }_p(V(p))$ is defined for every $p$.

The global exponential mapis smooth by smooth dependence of geodesics on initial data. Since $p\mapsto (p,V(p))$ is a smooth section of $TS$, their compositionis smooth.

For $0\le t\le 1$, defineCompleteness makes this a well-defined smooth homotopy, with ${\varphi }_0={\mathrm{id}}_S$ and ${\varphi }_1=\varphi$. By homotopy invariance of degree modulo two,This is the exponential displacement map on a compact surface.