# Space of divergence-free vector fields on a Riemannian manifold

Let $$(M,h)$$ be a smooth Riemannian manifold of dimension $$d\geq 1$$ with smooth metric. Set $$X:=\{A= \mbox{smooth vector field s.t. } div_h A=0 \}$$.

Then $$X$$ is an infinite dimensional vector space. How to prove this?

Comment: I am aware of the question

How to prove the space of divergence-free vector fields on a manifold is infinite dimensional?

but the proof therein does not convince me completely: I cannot see why one should be able to write locally $$dV_h= dx^1\cdots dx^d$$

• – Asaf Shachar Nov 14 '18 at 14:07