I'm attending a course in Symplectic Mechanics and I have some problems in understanding something written in my lecture notes.
We are in the following setting: let $Q$ be a manifold (of dimension $n$) and let $\Sigma$ be a connected submanifold of dimension $n-1$ in $Q$. We denote with $$j\colon\Sigma\hookrightarrow Q$$ the inclusion mapping. Suppose we are given a smooth function on $\Sigma$, $\sigma\colon\Sigma\to\mathbb{R}$ (which is understood to represent the initial or boundary data for our Cauchy problem - see below), and a Hamiltionian function $H\colon T^{*}Q\to \mathbb{R}$ (so $H$ is smooth too). We denote by $\pi_{Q}\colon T^{*}Q\to Q$ and by $\tau_{Q}\colon TQ\to Q$ the projections onto the base manifold (the map of the bundles).
Now start troubles: my professor defines $$L_{(\Sigma,\ \sigma)}:=\{p\in T^{*}Q:\ (d\sigma)(v)= p((Tj)(v)),\ \forall v\in T\Sigma,\ \pi_{Q}(p)=\tau_{Q}(v)\}.\qquad \left(\triangle\right)$$ Here $d\sigma\colon \Sigma\to T^{*}\Sigma$ associates to $q\in Q$ the map $d_{q}\sigma\colon T_{q}Q\to \mathbb{R}(\simeq T_{\sigma (q)}\mathbb{R})$ and $Tj\colon T\Sigma\to TQ$ is the map between the tangent bundles induced by the inclusion. (I also think that, in $\left(\triangle\right)$, $\pi_{Q}(p)=\tau_{Q}(v)$ is a slight abuse of notation for $\pi_{Q}(p)=\tau_{Q}((Tj)(v))$).
For my way of reasoning, the definition in $\left(\triangle\right)$ was not clear enough, so I asked my professor about it and, for what I was able to understand, $\left(\triangle\right)$ should be suitably written as:
$$L_{(\Sigma,\ \sigma)}:=\{(q,\ p)\in T^{*}Q:\ \forall (\lambda,\ v)(\ ((\lambda,\ v)\in T\Sigma\ \land q=\lambda )\implies (d\sigma)(v)= p((Tj)(v))\ )\}.$$ Well, if this is the right way of understanding this definition, if I'm not missing something, we get that if $q\in Q \setminus \Sigma$, then every $(q,p)\in T^{*}Q$ should belong to $L_{(\Sigma,\ \sigma)}$ because the hypotheses in the implication defining $L_{(\Sigma,\ \sigma)}$ is always false.
Now, there is another definition in my lecture notes. If $e\in \mathbb{R}$ is a regular value for $H$, a solution of the Geometric Cauchy Problem (for the Hamilton-Jacobi system associated to H) is a Lagrangian submanifold $\Lambda\subseteq H^{-1}(\{e\})$ such that
$$j^{*}(\Lambda\cap L_{(\Sigma,\ \sigma)})=(d\sigma)(\Sigma).\qquad (\square)$$
I basically can not understand what the lhs of this equality should be: I know pullbacks of vector fields and of differential forms, but I don't know what it means to do a pullback of a submanifold of the cotangent bundle. I can't give any concrete meaning to the lhs of $(\square)$, even trying to force it with some abuse of notations.
So my questions are:
Is my rephrasing of $(\triangle)$ meaningful?
What's the meaning of the lhs in $(\square)$ and of the equality itself?
Thanks in advance!
Edit: Just to be sure there is no confusion, the definition of a differential k-form for me is this one.