## Some challenging differential geometry problem

Given three maps: $i_{\sigma}:\sigma \rightarrow S\times T$, $\Pi_{S}:S\times T\rightarrow S$,and $\Pi_{T}:S\times T\rightarrow T$ where $\sigma$ is just a sub-manifold ($m-$dimension) of $S\times T$ and $i_{\sigma}$ is the inclusion map, and $\Pi_{S,T}$ being the projection maps.

**PROBLEM:** Given some smooth differential $m-$form $\chi$ on $S$ (dimension $m$) where $\chi (k)\ne 0$ for any $k \in S$, then $\exists$ $\sigma =\{(k,\Omega(k))\in S\times T: k\in S\}$ for some $\Omega:S\rightarrow T$ smooth **IF** $i_{\sigma}^{*}\Pi_{S}^{*}\chi$ is not zero and there is an bijection defined by the map $\Pi_{S}\circ i_{\sigma}.$

I do not know how to solve this problem from my notes, so I am asking for help online. Any help will be appreciated.