Suppose that $(M,g)$ is a Riemannain manifold $dV_g$ is its volume element and $X$ is a non-zero smooth vector field on $M$. If for every 1-form $\delta$ we have the following equality, $$\int_M \delta (X) dV_g=0.$$
What is the most important result can we deduce about $X$?
I tried with $$0=\int_M \delta (X) dV_g=\int_M <i_X g,\delta> dV_g$$ and these equalities hold for any $\delta$ if $X=0$. But the assumption specifies that $X\neq 0$. So I think the condition of problem never happens.