On Lagrangian embeddings of parallelizable manifolds (Q2855494)

In this paper, the authors study the problem of realizing an \(n\)-manifold \(M^n\) as a Lagrangian submanifold in the \(2n\)-dimensional Euclidean space \(\mathbb R^{2n}\) with a certain symplectic structure. To obtain a Lagrangian embedding of an \(n\)-manifold in \(\mathbb R^{2n}\) it is sufficient to embed its cotangent bundle in \(\mathbb R^{2n}\) and extend its symplectic structure to \(\mathbb R^{2n}\). By using Gromov's \(h\)-principle for symplectic structures on an open manifold and the Kervaire semi-characteristic, the authors prove that for any closed parallelizable \(n\)-manifold \(M^n\), if the dimension \(n\neq 7\), or if \(n=7\) and the Kervaire semi-characteristic \(\chi_{\frac12}(M^7)\) is zero, then \(M^n\) can be embedded in the Euclidean space \(\mathbb R^{2n}\) with a certain symplectic structure as a Lagrangian submanifold. It turns out that for \(n=2\), the only closed parallelizable \(2\)-manifold is the \(2\)-torus and for \(n=3\), any closed orientable \(3\)-manifold is parallelizable. There are infinitely many isotopy classes of embeddings of the \(2\)-torus in the \(4\)-dimensional Euclidean space. For \(n\geq 3\), there is a surjection from the set of isotopy classes of embeddings of \(M^n\) in the \(2n\)-dimensional Euclidean space \(\mathbb R^{2n}\) to the homology group \(H_1(M^n;\mathbb Z)\) if \(n\) is odd, and to \(H_1(M^n;\mathbb Z_2)\) if \(n\) is even. By the results of Gromov and Fukaya, the authors' result gives rise to symplectic structures of \(\mathbb R^{2n}\) \((n\geq 3)\) which are not conformally equivalent to open domains in standard ones.