Assume $M^n\subset (W^{2n},\omega)$ be $n$-dimensional manifold with smooth boundary $\partial M$, where $(W,\omega)$ be 2n-dimensional K"{a}hler manifold and boundary with contact type structure, which ensure that $\partial W$ have a contact 1 form induced from $\omega$.

My question is: How to understand the structure of $\partial T^*(M)$? and characterize its normal vector field in $T^*M$?Can it be related to the relative normal field of $\nu\in T_{\partial M}M$>?

What is relation between $\partial T^*(M)$ and jet bundle $T^*(\partial M)\times \mathbb{R}$ with standard contact 1 form, and when can $T^*(\partial M\times \mathbb{R})$ and $T^*(M)$ can be identitied?

moreover, what if $M$ a be Lagrangian submanifold?

Many thanks for any comments or examples.

Suppose that $M\subset (W^{2n},\omega)$ is an $n$-dimensional manifold with smooth boundary $\partial M$, where $(W,\omega)$ is a $2n$-dimensional Kähler manifold and boundary with contact type structure, which ensure that $\partial W$ has a contact 1-form induced by $\omega$.

My questions are:

1. How can we understand the structure of $\partial T^*(M)$ and characterize its normal vector field in $T^*M$? Can it be related to the relative normal field of $\nu\in T_{\partial M}M$?
2. What is the relation between $\partial T^*(M)$ and jet bundle $T^*(\partial M)\times \mathbb{R}$ with standard contact 1-form, and when can $T^*(\partial M\times \mathbb{R})$ and $T^*(M)$ be identified?
3. What if $M$ is a Lagrangian submanifold?

Many thanks for any comments or examples.