Let $X$ be a smooth manifold of dimension $n$, where smooth manifolds are understood to be Hausdorff and second countable, let $N \geq 1$, and let $\mathbb{R}^N_0 := \mathbb{R}^N \setminus \{0\}$. Let $\Omega \subset X \times \mathbb{R}^N_0$ be an open subset that is conic in the second variable, meaning that $(x,\theta) \in \Omega$ and $r > 0$ imply $(x,r\theta) \in \Omega$. Let

\begin{align*} \phi:\Omega \to \mathbb{R} \end{align*}

be a smooth function that is positively homogeneous of degree $1$ in $\theta$, so that $\phi(x,r\theta)=r\phi(x,\theta)$ for every $(x,\theta)\in\Omega$ and every $r>0$.

Define the critical set

\begin{align*} C_\phi := \{(x,\theta)\in\Omega : d_\theta\phi(x,\theta)=0\}. \end{align*}

Assume that $\phi$ is nondegenerate in the following sense: for every $(x,\theta)\in C_\phi$, the covectors

\begin{align*} d_{x,\theta}(\partial_{\theta_1}\phi)(x,\theta),\dots,d_{x,\theta}(\partial_{\theta_N}\phi)(x,\theta) \end{align*}

are linearly independent in $T_{(x,\theta)}^*\Omega$, and $d_x\phi(x,\theta)\neq 0$ in $T_x^*X$.

Define

\begin{align*} j_\phi:C_\phi \to T^*X\setminus 0,\quad (x,\theta)\mapsto (x,d_x\phi(x,\theta)). \end{align*}

Then $C_\phi$ is a smooth conic submanifold of $\Omega$ of dimension $n$, the map $j_\phi$ is a conic immersion, and

\begin{align*} \Lambda_\phi := j_\phi(C_\phi) \end{align*}

is an immersed conic Lagrangian submanifold of $T^*X\setminus 0$, with its immersed structure induced by $j_\phi$. If $C_\phi=\varnothing$, this conclusion is understood under the standard convention that the empty set is a smooth submanifold of any specified dimension and its image is an immersed Lagrangian submanifold vacuously. If, in addition, $j_\phi:C_\phi\to T^*X\setminus 0$ is an embedding, then $\Lambda_\phi$ is an embedded conic Lagrangian submanifold. In particular, this embeddedness conclusion holds if $j_\phi$ is a proper injective immersion.