Let $X$ and $Y$ be smooth manifolds, let $N \in \mathbb{N}$, and let $\Omega \subset X \times Y \times \mathbb{R}^{N}_{0}$ be open and conic in the $\theta$ variable. Let $\phi \in C^\infty(\Omega;\mathbb{R})$ be positively homogeneous of degree $1$ in $\theta$. Define
paragraph
admin
\begin{align*}
\Sigma_\phi := \{(x,y,\theta) \in \Omega : \partial_{\theta_j}\phi(x,y,\theta)=0 \text{ for every } 1 \leq j \leq N\}.
\end{align*}
latex_env
admin
Assume that, at every point of $\Sigma_\phi$, the covectors
with the immersed structure induced by $F_\phi$. Then $F_\phi$ is a conic Lagrangian immersion. Consequently $C_\phi$ is a conic immersed Lagrangian submanifold of $(T^*X\setminus 0)\times (T^*Y\setminus 0)$ equipped with the symplectic form $\pi_X^*\omega_X-\pi_Y^*\omega_Y$, where $\omega_X$ and $\omega_Y$ denote the canonical symplectic forms on $T^*X$ and $T^*Y$.