Question: If $\mu$ is a 1-form sufficiently $C^1$-close to the zero 1-form, then $$\{(p, \mu_p) ; p \in M, \mu_p \in T_p^*M \}\cong\text{Graph}\ f$$ for some diffeomorphism $f: M \to M$.
Attempt: Let $\sigma_0 : M \hookrightarrow T^*M $ the zero section (1-form) given by $\sigma_0 (p) = 0_p$. Consider $\mu: M \to T^*M$ a 1-form sufficently $C^1$-close to $\sigma_0$, that is, a section of the bundle $T^*M$, let $M' = \mu (M)$ be the image under $\mu$. Given $p \in M$, let $(q, 0_q) = \exp (p,\mu_p)$ be the point in $M_0 = \sigma_0 (M)$ given by the exponential map in a neighborhood of $M'$ seen as the zero section of the normal bundle $NM'$. Define $f : M \to M$ as the composition
where $\pi: T^*M \to M$ is the canonical bundle projection. Since $\pi \circ \mu = id_M$, $\pi \circ \sigma_0 = id_M$ and $\exp$ is a diffeomorphism, $f$ is smooth bijective with smooth inverse $\pi \circ \exp^{-1} \circ \ \sigma_0$.
Identifying $(p,\mu_p) \mapsto (p,f(p))$ gives the desired isomorphism.
- Is it correct? Is there anything to be improved in the argument?
