Timeline for Why is the standard contact structure on $\mathbb R^{2n+1}$ called "standard"?

8 events

when toggle format	what		by	license	comment
Dec 1, 2020 at 23:44	vote	accept	boink
Dec 1, 2020 at 23:43	vote	accept	boink
Dec 1, 2020 at 23:44
Dec 1, 2020 at 17:36	comment	added	Andy Sanders		Forgiving historical inaccuracies (and perhaps eliding constants), I would wager it has something to do with the restriction of the (canonical) Liouville one form on $T^{*}\mathbb{R}^{n}$ to the unit co-sphere bundle with respect to the Euclidean metric on $\mathbb{R}^{n}.$
Dec 1, 2020 at 9:13	answer	added	Marco Golla		timeline score: 4
Dec 1, 2020 at 9:03	answer	added	Thomas Rot		timeline score: 4
Dec 1, 2020 at 5:42	comment	added	Panagiotis Konstantis		Like in symplectic geometry, there is a Darboux theorem in contact geometry, which tells you that on a manifold with contact structure there is always a chart such that the given contact structure looks like the standard contact structure on $\mathbb R^n$.
Dec 1, 2020 at 4:56	comment	added	Deane Yang		Given a function $f: \mathbb{R}^n \rightarrow \mathbb{R}$, consider the graph of f and its gradient, $G = \{ (x_1, \partial_1f, \dots, x_n,\partial_nf, f) \}$. The standard contact form $\alpha$ vanishes when restricted to this graph. Conversely, any $n$-dimensional submanifold on which $\alpha$ vanishes but the $n$-form $dx_1 \wedge\cdots\wedge dx_n \ne 0$ at every point is the graph of a function and its gradient. This is how the concept of a contact structure first arose.
Dec 1, 2020 at 4:51	history	asked	boink	CC BY-SA 4.0