dg.differential geometry - A question about a one-form on Riemannian manifold

Assuming the dimension of $M$ is at least 2 (otherwise it's false), you can do the following. Let $p_1,p_2,dots$ be isolated points where $X$ does not vanish but where you want $omega$ to vanish. In a neighborhood $U_i$ of each $p_i$, there are coordinates $(x^1,dots,x^n)$ centered at $p_i$ on which $X$ has the coordinate representation $X = partial/partial x^1$. In each $U_i$, let $omega_i = dx^2 + |x|^2 dx^1$. Then let $U_0$ be the complement of {$p_1,p_2,dots$}, and let $omega_0=X^flat$ (the 1-form dual to $X$ via the metric). Let {$phi_0,phi_i$} be a partition of unity subordinate to the cover {$U_0,U_i$}, and let $omega = sum_{ige 0}phi_iomega_i$. The fact that $omega_i(X)>0$ at points other than $p_i$ and zeros of $X$ ensures that $omega(X)$ vanishes only at such points.

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