Suppose $M$ and $N$ are symplectic manifolds, $N$ is compact, $\dim_{\mathbb R}(N) \leq \dim_{\mathbb R}(M) -4$. Suppose there are embeddings $f_i:N \to M$, $i=0,1$ such that $f_i^*\omega_M$ is non-degenerate. Further there is a smooth path of embeddings connecting $f_0$ to $f_1$. That is, $h:[0,1] \times N \to M$ is such that $h_0=f_0$, $h_1=f_1$ and each $h_t$ is an embedding. Is it possible to perturb this path so that for each embedding on the path, the pull-back of $\omega_M$ is non-degenerate on $N$?

By Gromov's h-principle (see Theorem 12.1.1 in Eliashberg's h-principle book), any point $h_t$ in the path is $C^0$-close to a map $h_t'$ so that the pullback $(h_t')^*\omega_M$ is non-degenerate. I do not understand h-principles very well, I am wondering if this result can be used to perturb the whole path.



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