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Question

Math 6510 • Differentiable Manifolds

Summer 2016 • Question 5

Question

Math 6510 • Differentiable Manifolds Summer 2016 • Question 5

Let \(a:S^2\to S^2\) be the antipodal map on the 2-sphere: \(a(x)=-x\), and let \[ p:S^2\to \mathbb {R}\mathrm {P}^2=S^2/x\sim -x,\qquad p(x)=[x] \] be the projection to the projective plane. Let \(\omega \) be a 2-form on \(\mathbb {R}\mathrm {P}^2\).

Prove that \(\int _{S^2}p^*\omega =0\). Hint: \(p^*\omega \) is \(a\)-invariant.
Prove that there is a 1-form \(\eta \) on \(\mathbb {R}\mathrm {P}^2\) such that \(\omega =d\eta \). You are allowed to use the fact that if \(\zeta \) is a 2-form on \(S^2\) and \(\int _{S^2}\zeta =0\) then \(\zeta \) is exact, but you are not allowed to use de Rham’s theorem.

Let \(a:S^2\to S^2\) be the antipodal map on the 2-sphere: \(a(x)=-x\), and
let
\[
  p:S^2\to\RP^2=S^2/x\sim -x,\qquad p(x)=[x]
\]
be the projection to the projective plane. Let \(\omega\) be a 2-form on \(\RP^2\).
\begin{enumerate}
\item[(a)] Prove that \(\int_{S^2}p^*\omega=0\). Hint: \(p^*\omega\) is
\(a\)-invariant.
\item[(b)] Prove that there is a 1-form \(\eta\) on \(\RP^2\) such that
\(\omega=d\eta\). You are allowed to use the fact that if \(\zeta\) is a 2-form on
\(S^2\) and \(\int_{S^2}\zeta=0\) then \(\zeta\) is exact, but you are not allowed to
use de Rham's theorem.
\end{enumerate}

Topic Tags

Fixed Point Theorems and Borsuk-Ulam Real Projective Spaces Surface Topology and Surface Groups de Rham Cohomology