Let $M,N$ be smooth oriented Riemannian manifolds; $M$ closed and $N$ with non-empty boundary. Let $f:M \to N$ be smooth and suppose the image of $f$ intersects $\partial N$.
Let $$W:=\{V \in \Gamma(f^*TN)\, | \, \text{There exist a variation }\, f_t \text{ with variation field } V \}.$$
(By a variation we mean a smooth family $f_t$, such that $f_0=f,\frac{\partial f_t}{\partial t}|_{t=0}=V$).
Question:
Let $A \in \Gamma(f^*TN)$ and suppose $\langle A,V \rangle=0$ for every $V \in W$. Is it true that $A=0$?
(The inner product $\langle A,V \rangle$ is the integral product of sections of $f^*TN$ induced by the metrics on $M,N$).
The point is that $W$ is constrained, it is a strict subset of $\Gamma(f^*TN)$: For every $p \in M$, such that $f(p) \in \partial N$, $V(p)$ must be tangent to $\partial N$ or "inward-pointing" (it cannot point outside). This is clearly a necessary condition for a variation to exist.
A subquestion: Is this condition sufficient for the existence of a variation?
Since $W \subsetneq \Gamma(f^*TN)$ we cannot use directly the density of the smooth sections to conclude $A=0$.
Of course, the motivation is calculus of variations. If there is some reference which treats this issue I would be happy to know.