# Lagrangian Torus of a symplectic manifold

Suppose $$(M,\omega)$$ is a $$2n$$-dimensional symplectic manifold. Consider the $$n$$-torus $$\mathbb{T}^n=\mathbb{R}^n/\mathbb{Z}^n.$$ Then $$\mathbb{T}^n$$ is a Lagrangian submanifold of the symplectic $$2n$$-torus $$\mathbb{T}^{2n}$$, equipped with the unique symplectic form that pulls back to the canonical symplectic form on $$\mathbb{R} ^{2n}$$. Now, how does it follow that there exists a Lagrangian submanifold of $$M$$ that is diffeomorphic to $$\mathbb{T}^{n}$$?

• How is $M$ related to the torus? Are you asking if any symplectic $2n$-manifold admits such a submanifold? Commented Jun 25, 2020 at 22:03

The fact that $$\mathbb T^n$$ is a Lagrangian submanifold of $$\mathbb T^{2n}$$ plays no role. Indeed, by Darboux's theorem, for any symplectic manifold $$(M^{2n}, \omega)$$ and for any $$x\in M$$, there is $$U_0 \subset \mathbb R^{2n}$$ and a local coordinates chart $$\phi : U_0 \to U$$ so that $$\phi^* \omega = \omega_{std}$$ and $$x\in U$$, where $$\omega_{std}$$ is the standard symplectic structure on $$\mathbb R^{2n}$$. Since it is easy to find Lagnrangian in $$(U_0, \omega_{std})$$ diffeomorphic to $$\mathbb T^n$$: choose $$p =(p_1, \cdots, p_n)\in U_0$$ and $$r_1, \cdots, r_n >0$$ small enough, then $$\mathbb S^1 (p_1) \times \cdots \times \mathbb S^1 (p_n)$$ is such an example.