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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Spring 2019 • Question 4

Question

Math 6510 • Differentiable Manifolds Spring 2019 • Question 4

Let \(a:S^2\to S^2\) be the antipodal map \(a(x)=-x\) 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 two form on \(\mathbb {R}\mathrm {P}^2\).

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

Raw LaTeX

Let \(a:S^2\to S^2\) be the antipodal map \(a(x)=-x\) 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 two 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 one form \(\eta\) on \(\RP^2\) such that
\(d\eta=\omega\). You are allowed to use the fact that if \(\beta\) is a two-form on
\(S^2\) and \(\int_{S^2}\beta=0\) then \(\beta\) is exact, but you are not allowed to
use the de Rham Theorem.
\item[(c)] Prove that \(\RP^2\) is not orientable.
\end{enumerate}

Topic Tags

Fixed Point Theorems and Borsuk-Ulam Orientability and Volume Forms Real Projective Spaces de Rham Cohomology