I have trouble understanding the following example:
1)First: How to understand the definition? $T_{(x,y)}(M \times M)= T_xM \times T_yM$. Let $(v,w), (\tilde{v},\tilde{w})\in T_{(x,y)}(M \times M)$ Then $(- \omega) \times \omega( (v,w), (\tilde{v},\tilde{w}))=- \omega_x(v, \tilde{v}) \cdot \omega_y(w, \tilde{w})$. Is that correct?
2)Then, by $M \times pt$ is meant some point $p$ and $L:=M \times \{p\}$ Is that correct? Then I don't see how this is symplectic, since $T L= T_M \times \{0\}$.
3) Also I don't see how to prove, that the diagonal $\Delta$ is Lagrangian since I don't know how to describe $T_(x,x) \Delta$.
Thanks for any help on my questions!
The symplectic form on the product is defined by $\Omega((u,v);(u',v'))=-\omega(u,u')+\omega(v,v')$.
This implies that the restriction of $\Omega$ to $M\times \{pt\}$ is $-\omega$ and is symplectic.
A tangent vector of $\{x,x\}$ is $(u,u),u\in T_xM$ this implies that $\Omega((u,u);(v,v)=-\omega(u,v)+\omega(u,v)=0$ and the diagonal is Lagrangian because its dimension is the half of the dimension of $M\times M$.