Definition of the pullback of $L=Im(dS)$ and related questions on defintions

Question

Def: We call a phase function $S: \mathbb{R}^n \rightarrow \mathbb{R}$ admissible if it satisfies the Hamilton-Jacobi equation. The image $L=Im(dS)$ of the differential of an admissible phase function $S$ characterzied by three geometric properties:

1. $L$ is an $n$-dimensional submanifold of $H^{-1}(E)$.
2. The pullback to $L$ of the form $\alpha_n = \sum_j p_j dq_j$ on $\mathbb{R}^{2n}$ is exact.
3. The restriction of the canonical projection $\pi: T^*\mathbb{R}^n \rightarrow \mathbb{R}^n$ to $L$ induces a diffeomorphism $L \simeq \mathbb{R}^n$. In other words, $L$ is projectable.

I have the following questions:

1. What is the definition of the pullback in this case? I saw many definitions on wikipedia and I'm not sure which one is supposed to be applied here.
2. What is $T$ in $T^*\mathbb{R}$? It is defined so that the phase space is $\mathbb{R}^2 \simeq \mathbb{R}$, which is not so clear.

Note: This definition comes from "Lectures on the Geometry of Quantization", by Bates & Weinstein.

Answer 1

1. "Pullback to $L$" here means the pullback by the inclusion map $i_L:L\to T^*\mathbb{R}^n$. Since $L$ is characterised by the equation $p = dS$, i.e. $p_i = \frac{\partial S}{\partial q^i}$, we have that $$ i_L^*\alpha_n = \frac{\partial S}{\partial q^i}dq^i = dS $$ so it's exact.

2. $T^*\mathbb{R}$ is the cotangent bundle of $\mathbb{R}$, which is the set of cotangent vectors over points of $\mathbb{R}$. For any $x\in\mathbb{R}$, the cotangent space $T^*_x\mathbb{R}$ at $x$ is the dual space to the tangent space $T_x\mathbb{R}$, and so is isomorphic to $\mathbb{R}$. Therefore $T^*\mathbb{R} = \sqcup_{x\in\mathbb{R}}T_x^*\mathbb{R} \simeq \mathbb{R}\times\mathbb{R}$.