Math 6510 • Differentiable Manifolds

In this problem you are allowed to use the fact that every closed 1-form on the 2-sphere \(S^2\) is exact and every 2-form \(\omega \) on \(S^2\) such that \(\int _{S^2}\omega =0\) is exact. You are also allowed to use that \(\int _{S^2} a^*(\omega )=-\int _{S^2}\omega \) for every 2-form \(\omega \) on \(S^2\), where \(a:S^2\to S^2\) is the antipodal map. Consider the map \(p:S^2\to \mathbb {R}P^2\) that identifies the antipodal points. Show that all closed 1- and 2-forms on \(\mathbb {R}P^2\) are exact by considering their pullbacks to \(S^{2}\) and applying the above facts.