Immersions in a manifold with a pair of symplectic forms (Q719903)

The paper under review offers several interesting results in connection with the problem of finding an immersion \(f: M\to N\) such that \(f^*(\sigma_j)=\omega_j\) for \(j=1,2\), where \((\sigma_1,\sigma_2)\) is a pair of symplectic forms on the smooth manifold \(N\), while \((\omega_1,\omega_2)\) is a pair of closed 2-forms on the manifold \(M\). We describe here only one of these results. Let us assume that \(N={\mathbb R}^{4q}\) endowed with the symplectic forms \(\sigma_1=\sum_{k=1}^{2q} dx_k\wedge dy_k\) and \(\sigma_2=\sum_{k=1}^q(dx_{2k-1}\wedge dy_{2k}-dx_{2k}\wedge dy_{2k-1})\). If \(M\) is a closed manifold endowed with the exact 2-forms \(\omega_1\) and \(\omega_2\), and \(2q\geq 3\dim M\) and the natural number \(q\) is even, then there exists a \((\sigma_1,\sigma_2)\)-regular immersion \(f: M\to{\mathbb R}^{4q}\) satisfying the conditions \(f^*(\sigma_j)=\omega_j\) for \(j=1,2\). This and related results are obtained by using the analytic technique and the sheaf technique in the theory of \(h\)-principle; see the book by \textit{M. Gromov} [Partial differential relations. Berlin etc.: Springer-Verlag (1986; Zbl 0651.53001)].