Back to Real Math: Symplectic Structure on Co-tangent Bundles

Here is a beautiful construction that I lost, and had to reproduce. I found hints on the web, but nowhere a clear construction, so here it is: Let M be a differential manifold. Let T^{*}M, TM be the tangent and co-tangent bundle, respectively. Now let TT^{*}M be the tangent bundle of the co-tangent bundle. The bundle map T^{*}M->M is differentiable, and so extends to a map F:TT^{*}M->TM. Also, there is the bundle map G:TT^{*}M->T^{*}M. Now let v be in TT^{*}_{m}M, F(v) is in T_{m}M and G(v) is in T^{*}_{m}M, which can be canonically identified with T_{m}M^{*}. So G(v)(F(v)) is well defined. Call this \(v). \ is well defined and only depends on local differential properties. Over U, an open subset of R^n, TT^{*}U can be identified with U X R^n X R^n, and \(u, v, w)=(v,w), the inner product. So, \ is a differential function from TT^{*}M->R which is linear above every point — a differential form on T^{*}M. Thus, d\ is an exact, and so closed, differential 2-form. Note that on T^{*}U, which can be canonically identified with a subset of R^n X R^n, it is equal to the canonical symplectic form, and since being non-degenerate is a local property, d\ is a non-degenerate closed 2-form — a symplectic form.

Conclusion: for every manifold M, T^{*}M has a canonical symplectic structure, given by d\.

(\ is apparently called the Liouville form, but I haven’t found a nice construction of it online.)

EDIT: Changed “non-trivial” to “non-degenerate”, as suggested by Professor Karshon. She also mentioned Ana Cannas da Silva’s book as a reference, though the site was down when I tried to confirm the link.

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One Response to Back to Real Math: Symplectic Structure on Co-tangent Bundles

didn’t even try to read beyond “manifold”, just wanted to make sure you ARE aware that wordpress.com allows you to type equations in TeX like a pro…