Let $n \in \mathbb{N}$ with $n \ge 1$, let $X$ be a smooth finite-dimensional manifold of dimension $n$, and for each $x \in X$ let $T_x^*X$ denote the cotangent space at $x$. Let $\mathcal{L}^n$ denote $n$-dimensional [Lebesgue measure](/page/Lebesgue%20Measure) on $\mathbb{R}^n$. Let the nonzero cotangent bundle be

\begin{align*} T^*X \setminus 0 := \{(x,\xi) \in T^*X : \xi \in T_x^*X \setminus \{0\}\}. \end{align*}

Let $u \in \mathcal{D}'(X)$ be a distribution on $X$. Let $WF(u) \subset T^*X \setminus 0$ denote the smooth wave front set of $u$, defined in coordinate charts by the local conic rapid-decay condition for Fourier transforms of compactly supported coordinate localizations, where Fourier transforms in local coordinates are taken with respect to $\mathcal{L}^n$. Let $\operatorname{sing\,supp}(u) \subset X$ denote the singular support of $u$, defined as the complement of the largest open subset of $X$ on which $u$ is represented by a smooth function. Let

\begin{align*} \pi:T^*X \setminus 0 \to X \end{align*}

be the cotangent bundle base projection, defined by $\pi(x,\xi)=x$. Then

\begin{align*} \pi(WF(u))=\operatorname{sing\,supp}(u). \end{align*}