= 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 $T_pS$. In particular, $\exp_p(V(p))$ is defined for every $p$.
The global exponential map
$$
\operatorname{Exp}:TS\to S,
\qquad (p,v)\mapsto\exp_p(v),
$$
is smooth by smooth dependence of geodesics on initial data. Since $p\mapsto(p,V(p))$ is a smooth section of $TS$, their composition
$$
\phi(p)=\operatorname{Exp}(p,V(p))
$$
is smooth.
For $0\leq t\leq1$, define
$$
\phi_t(p)=\exp_p(tV(p)).
$$
Completeness makes this a well-defined smooth homotopy, with $\phi_0=\operatorname{id}_S$ and $\phi_1=\phi$. By homotopy invariance of <degree modulo two>,
$$
\deg_2(\phi)=\deg_2(\operatorname{id}_S)=1.
$$
This is the <exponential displacement map on a compact surface>.
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