a contact structure
Symplectic geometry lives in even dimensions; its odd-dimensional cousin is contact geometry. Picture, at every point of an odd-dimensional space, a hyperplane — a tilted flat slab one dimension below the whole space — and let these slabs twist so violently from point to point that no surface can ever stay tangent to all of them. That impossibly twisted field of hyperplanes is a contact structure. The boundary of a symplectic phase space, a constant-energy surface, and the space of contact elements of any manifold all carry one naturally.
Precisely, a contact structure on a (2n+1)-dimensional manifold M is a hyperplane field (a smooth choice of 2n-dimensional subspace xi_p in each tangent space T_p M) that is maximally non-integrable. Locally xi = ker(alpha) for a 1-form alpha called a contact form, and the maximal non-integrability condition is alpha ^ (d alpha)^n is nowhere zero — a top-degree form that never vanishes. This is the exact opposite of the Frobenius integrability condition alpha ^ d alpha = 0 that would make xi tangent to a foliation; here the twisting is as strong as possible. The contact form is not unique (multiplying alpha by any nonzero function f gives the same kernel xi), so the structure is the hyperplane field xi, with alpha an auxiliary choice. The standard model is R^{2n+1} with coordinates (z, x_1, y_1, ..., x_n, y_n) and alpha = dz - sum y_i dx_i.
Why it matters: contact manifolds are the natural odd-dimensional boundaries and energy levels of symplectic manifolds (the symplectization M x R of a contact manifold is symplectic), and contact geometry has its own Darboux theorem — all contact structures are locally identical to the standard model, so again there are no local invariants. The global theory is rich and surprising: in dimension 3, contact structures split into 'tight' and 'overtwisted' types with completely different rigidity (Eliashberg's classification), and Reeb dynamics on contact manifolds is where the Weinstein conjecture on closed orbits lives. A common confusion is between a contact structure (the hyperplane field xi, the geometric object) and a contact form (a specific alpha cutting it out); the form carries extra data — most importantly it pins down a Reeb vector field — that the bare structure does not.
On R^3 with coordinates (x, y, z) the standard contact form is alpha = dz - y dx, so the contact planes xi = ker(alpha) are spanned by partial/partial y and partial/partial x + y partial/partial z. Check non-integrability: d alpha = -dy ^ dx, so alpha ^ d alpha = (dz - y dx) ^ (-dy ^ dx) = dz ^ dx ^ dy (up to sign), which is the nonzero volume form. The planes twist a full quarter-turn as you move along the y-axis — no surface can stay tangent to them.
Standard R^3 contact structure alpha = dz - y dx: planes twist so alpha ^ d alpha never vanishes.
Maximal non-integrability is the opposite extreme from a foliation. Distinguish the contact structure xi (the hyperplane field, well-defined up to nothing) from a contact form alpha (one of infinitely many functions f times alpha with the same kernel); only the form determines a Reeb field and a volume.