Let $M$ be a smooth n dimensional manifold. Is there an smooth embedding $f:M \to \mathbb{R}^{2n}$ whose image is a Lagrangian submanifold of $\mathbb{R}^{2n}$?



A lagrangian version of the Withney theorem
No: being a Lagrangian submanifold of $\mathbb{R}^{2n}$ imposes strong conditions on $M$. If $M$ is embedded in $\mathbb{R}^{2n}$, the bundle $T\_M\oplus N$ (normal bundle) is trivial; if $M$ is Lagrangian, the symplectic form induces an isomorphism $N\cong T\_M^\*$. Thus $T\_M\oplus T\_M$ is trivial; this implies for instance that the Pontryagin classes of $M$ are trivial in $H^\*(M,\mathbb{Q})$. So e.g. $\mathbb{CP}^2$ cannot be embedded as a Lagrangian submanifold of $\mathbb{R}^{8}$.


