 Symplectic plane

The standard symplectic plane is ${\mathbb{R}}^2$ with the closed nondegenerate two-form $\omega =dq\wedge dp$. For tangent vectors $u=(a,b)$ and $v=(c,d)$,their signed parallelogram area. This alternating pairing is not a positive-definite inner product: every vector pairs to zero with itself.

A linear map $A$ preserves the form exactly when $\det A=1$, because $\omega (Au,Av)=\det (A)\omega (u,v)$. Thus $\mathrm{Sp}(2,\mathbb{R})={\mathrm{SL}}_2(\mathbb{R})$. A nonlinear example is the shear $(q,p)\mapsto (q,p+q^2)$, whose pullback form is $dq\wedge (dp+2qdq)=dq\wedge dp$.

Every smooth embedded curve in the plane is Lagrangian: the restricted two-form vanishes on its one-dimensional tangent spaces, which have half the ambient dimension. The symplectic plane is therefore the simplest model of symplectic area preservation and Lagrangian geometry.