I want to describe submanifolds of $T^4:=\mathbb{R}^4/\mathbb{Z}^4$ which are of the type symplectic or lagrangian or isotropic or coisotropic (using the standard symplectic structure $dx_1 \wedge d y_1+dx_2 \wedge d y_2$.
My problem is, that, except guessing, I don't see a method of finding these.
For the standard euclidean product, in 2 or 3 dimensions I can imagine how the tangent space of a point looks like and how its orthogonal complement must look like (using the "graphical interpretation" of orthogonal). But I don't know how to visualise a symplectic product. Does somebody have some hints on that or general hints how to solve this?