- #1
RedTachyon
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Homework Statement
Let ## (M, \omega_M) ## be a symplectic manifold, ## C \subset M ## a submanifold, ## f: C \to \mathbb{R} ## a smooth function. Show that ## L = \{ p \in T^* M: \pi_M(p) \in C, \forall v \in TC <p, v> = <df, v> \} ## is a langrangian submanifold. In other words, you have to show that ## \omega |_L = 0 ## and ## dim\ L = \frac{1}{2} dim\ M ##
Homework Equations
## \alpha: M \to T^*M##
##\alpha^*\theta_M = \alpha## where ## \theta_M ## is Liouville's form ( ## d\theta_M = \omega_M ## )
The Attempt at a Solution
I'm trying to start with decomposing ## p \in L ## like that: ## p = df + \varphi ##, where ## \varphi \in TC^\circ ##, since ## p ## can differ from ## df ## only by ## \varphi:\ <\varphi,v> = 0\ \forall v \in TC ##. But I think there could be several possible ## \varphi ## for each point? There should be, since we know nothing about ## dim\ C##. I also tried doing something with the pullback ## P^*\theta_M ## where P would be a one-form that takes on the value ## p ## at ## \pi_M(p) ## but I can't really get any results.
Thanks in advance for any help!