Suppose you have a manifold $M$ with a volume form $\omega$. Let $f\in C^\infty(M)$ with a regular value $0$. Consider the codimension 1 submanifold $\Sigma=f^{-1}(\{0\})$. Intuitively, one could define the volume form $\omega_{\Sigma}=\omega\delta(f)$ on $\Sigma$. Is there a more geometric way of understanding this measure? For example, is there a natural way of writing $\omega_\Sigma=\iota_X\omega$, for some vector field $X\in\mathfrak{X}(M)$.
This sort of thing appears in the microcanonical analysis of continuous systems. For example, the measure appearing in the treatment of the classical harmonic oscillator https://physics.stackexchange.com/questions/406972/harmonic-oscillator-in-microcanonical-ensemble. In there one has $$\text{d}q\text{d}p\delta\left({\frac{p^2}{2}+\frac{q^2}{2}-E}\right)=\frac{2\text{d}q}{\sqrt{2E-q^2}},$$ where the $q$ on the right-hand side is actually the pullback of the coordinate $q$ on phase space $\mathbb{R}^2$ to the circle centered at the origin of radius $\sqrt{2E}$.
Choose a (local) vector field $X$ such that $X(f)=1$. The interior product $\iota_X\omega$ will depend on the choice of $X$, but its pullback to $\Sigma$ will not.
This volume form is related to the induced measure on the "infinitesimally fattening of $\Sigma$" $f^{-1}([0,\epsilon))$. Each such vector field $X$ defines a diffeomorphism $\varphi:\Sigma\times[0,\epsilon)\to f^{-1}([0,\epsilon))$ where the coordinate on the second factor is exactly $f$ (the flowout of $\Sigma$ along $X$). We can define a measure $\mu$ on $\Sigma$ by $$ \mu(U)=\lim_{\delta\to 0}\frac{1}{\delta}\mu_\omega(\varphi(U\times[0,\delta))) $$ Where $U\subseteq\Sigma$ is open and $\mu_\omega$ is the measure on $M$ induced by $\omega$. It turns out this limit does not depend on the choice of $X$, and is equivalent to the measure induced by $\iota_X\omega|_{\Sigma}$.