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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.

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  • $\begingroup$ Wouldn't $\partial T^*(M)$ be the same as $T^*(M)|_{\partial M}$ and there is a canonical projection from $T^*(M)|_{\partial M}$ to $T^*(\partial M)$ by restricting? $\endgroup$
    – Michael Bächtold
    Commented Jun 14, 2018 at 14:21
  • $\begingroup$ Thanks for comments. You mean view $\partial T^*(M)$ as $T^*(M)|_{\partial M}$ bundle over $\partial M$ with same $n$ dimensional fiber as $T^*(M)$ ? Is the normal vector field of $T^*(M)|_{\partial M}$ be induced from $\nu\in T_{\partial M}M$? say, for a section $s\in T^*(M)$ of cotangent bundle, is the normal vector field at point $(x,s(x))\in T^*(M)|_{\partial M} $( here, $x\in\partial M$) in $T^*(M)$ be $(\nu_x,\nu_x \lrcorner \nabla s)$? $\endgroup$
    – John Sung
    Commented Jun 14, 2018 at 15:34
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    $\begingroup$ Yes, for your first question. I don’t quite understand your second question. What do you mean with the normal vector field? There is a canonical 1 dimensional subspace of $T^*M|_{\partial M}$ consisting of those covectors that vanish when paired with tangent vectors of the boundary. A generator of this subspace might be called a normal covector. $\endgroup$
    – Michael Bächtold
    Commented Jun 14, 2018 at 17:03
  • $\begingroup$ I wonder whether the normal covector of $T^*M_{\partial M}$ (view it as submanifold and boundary of $T^*M$ ) has some relation with the normal covector of $\partial M$ ? $\endgroup$
    – John Sung
    Commented Jun 15, 2018 at 8:37

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