What does $w(V_p)$ nowhere zero mean?

Question

What does a differential $k$-form being nowhere $0$ actually mean? Does that mean that the tangent vectors it consumes and converts to a number never becomes $0?$ For instance for some tangent vector of a manifold $M$, let $V_p$ be a tangent vector at point $p$ on $M, V_p\in T_pM$(tangent space at point $p$ on $ M)$, for some given $k$-form on the cotangent space of $ M,$ does the $k$-form $w$ on $M$ nowhere being $0$ imply $w(V_p)\ne 0$ for every $V_p\in T_pM?$

Answer 1

I am not sure what nowhere $0$ means but my guess is that it means that the differential form is non vanishing. We say a differential form $\omega$ is nonvanishing if, for every $p\in M$, $\omega$ is not the zero mapping. That is to say, for every $p\in M$, there is at least one tangent vector $V_p\in T_pM$ such that $\omega(V_p)\neq 0$. It doesn't mean that $\omega(V_p)\neq 0$ for all $V_p\in T_pM$, there might be some tangent vectors $U_p$ for which $\omega(U_p)=0$. It just means that $\omega$ is not identically $0$ (i.e. nonzero in a mapping sense) and so there must be at least one tangent vector where $\omega(V_p)\neq 0$.