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# On the de Rham cohomology of 1-froms1-forms in cotangent bundle.

5 edited grammar, changed "cohomology group" to "cohomology class"

We know that a cotangent bundle $T^\star M$ has a canonical symplectic form and $M$ is a nature natural Lagrangian submanifold of it. A well known result is that any submanifold $X={(p,f(p)): p\in M}$, where $f$ is a closed one form is Lagrangian. Denote by $[f]$ the de Rham cohomology group class of $f$. Assume that we flow $X$ in a Hamiltonian direction to $Y$, then $Y$ will be a Lagrangian submanifold of $T^\star M$. My question is can we write it as $Y={(p,g(p)): p\in M}$ for some closed one form $g$.? If so, do we have $[g]=[f]?$ Thanks in advance!

4 I Edited the question according to the comments of Serge and Tagging. Thanks them!

# On the HodgehomologydeRhamcohomology of 1-froms in cotangent bundle.

We know that a cotangent bundle $T^\star M$ has a canonical symplectic form and $M$ is a nature Lagrangian submanifold of it. A well known result is that any submanifold $X={(p,f(p)): p\in M}$, where $f$ is a closed one form is Lagrangian. Denote by $[f]$ the Hodge homology de Rham cohomology group of $f$. Assume that we flow $X$ in a Hamiltonian direction to $Y$, then $Y$ will be a Lagrangian submanifold of $T^\star M$. My question is can we write it as $Y={(p,g(p)): p\in M}$ for some closed one form $g$.? If so, do we have $[g]=[f]?$ Thanks in advance!Edit: I edit the question according to Serge's comments.

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