We consider an $n$-dimensional Riemannian manifold $M$ (not necessarily compact) without boundary. Following the paper ''Heat semigroup and functions of bounded variation on Riemannian manifolds'' by Miranda et al., one can define the variation of a function $u \in L^1(M)$ as a measure given on $M$ by
$$ |D u|(M):=\sup \left\{\int_M u \operatorname{div} \omega \mathrm{~d} V_g: \omega \in \Gamma_c(T^* M),|\omega(x)|_g \leqslant 1 \text { for all } x \in M\right\}, $$
where $\Gamma_c(T^* M)$ denotes the space of all compactly supported $C^{\infty}$ vector fields. We say that $u$ is of bounded variation and write $u \in B V(M)$, if $|D u|(M)<\infty$. Taking a measurable set $E \subset M$, we denote the perimeter of $E$ in a Borel set $A$ by
$$ P(E, A)=\left|D 1_E\right|(A), $$ where $1_E$ is the characteristic function of $E$. Now, we recall the coarea formula for BV functions in the Euclidean case ($\Omega$ is an open set in $\mathbb{R}^n$), which says that for any $u \in \operatorname{BV}(\Omega)$, denoting $E_t:=\{x \in \Omega: u(x)>t\}, t \in \mathbb{R}$, it holds:
$$ |D u|(\Omega)=\int_{-\infty}^{\infty} P\left(E_t, \Omega\right) d t. $$
Is the same true also in the manifold case? Can you provide a reference about that?
