Geometry of (contact) manifolds

foliations; distributions; integrability (Frobenius theorem); contact structures

some symplectic linear algebra: symplectic forms and symplectic vector spaces; $\omega\in\Lambda^2(V^{2n})^\ast$ is symplectic iff $\omega^n\neq 0$; isotropic and Lagrangian subspaces of a symplectic vector space

coorientability of hyperplane distributions: if $\xi^{n-1}\subset TM^n$, then $\xi=\ker\alpha$ for some $\alpha\in\Omega^1M$ iff the line bundle $TM/\xi$ is orientable

$\alpha\in\Omega M^{2n+1}$ everywhere non-zero. Then $\ker\alpha$ is contact iff $\alpha(d\alpha)^n$ is a volume form

contact forms of coorientable contact structures

Cartan's magic formula: $\mathcal{L}_X=X\lrcorner d+d X\lrcorner$

the standard contact form on $(\mathbb{R}^{2n+1},x_1,y_1,\dots,x_n,y_n,z)$: $\alpha_{std}=dz-\sum y_idx_i$

contactomorphisms, contact isotopies, contact vector fields; "contact vector fields are the tangent vectors of the group of contactomorphisms at the identity"; contact vector fields on $(M,\xi)$ and sections of $TM/\xi$ "are the same" (as vector spaces of course); isotopy extension; if $(M,\ker\alpha)$ is contact, then $\alpha$ induces an iso between $TM/\xi$ and the trivial line bundle, and hence between contact vector fields and functions. The Reeb vector field of $\alpha$ is then the contact vector field corresponding to $1$

contact conditions in terms of contact forms: $\phi:(M,\ker\alpha)\rightarrow (M,\ker\alpha)$ is contact iff $\phi^\ast\alpha=f\alpha$ for some everywhere non zero $f\in C^\infty M$; $X\in\Gamma TM$ is contact iff $\mathcal{L}_X\alpha\in C^\infty(M)\alpha$

Gray's stability theorem ("contact structures have no deformation theory", or "the space of contact structures modulo diffeomorphisms isotopic to the identity is discrete"); Darboux theorem ($(\mathbb{R}^{2n+1},\alpha_{std})$ is a universal local model, contact manifolds have no local invariants)