# Open sets of symplectic manifolds

Suppose I have a symplectic manifold $(\mathcal{M}, \omega)$. Does it hold that any open subset of $(\mathcal{M}, \omega)$ is a symplectic submanifold?

The statement trivially holds for smooth manifold but in the symplectic setting I am not sure.

-

## 1 Answer

It trivially holds for symplectic manifolds as well. Let $(M^{2n}, \omega)$ be a symplectic $2n$-manifold and $X \subset M$ be an open subset. Then we know that $X$ is a $2n$-dimensional smooth submanifold of $M$. This implies that for all $x \in X$, $T_x M = T_x X$ (a $2n$-dimensional subspace of a $2n$-dimensional vector space is the whole vector space). Therefore since $(T_x M, \omega_x)$ is a symplectic vector space by the fact that $(M,\omega)$ is a symplectic manifold, $(T_x X, \omega_x)$ is a symplectic vector space as well. Therefore $X$ is a symplectic submanifold of $(M,\omega)$.

-
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 17, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9235660433769226, "perplexity_flag": "head"}