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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 1999 • Question 10

Question

Math 6510 • Differentiable Manifolds Fall 1999 • Question 10

Let \(\omega \) be a smooth closed \(p\)-form on \(S^n\) with \(p>0\). Show that there is a smooth \((p-1)\)-form \(\alpha \) so that \(d\alpha =\omega \) if \(p<n\) but that this need not be the case when \(p=n\). You are allowed to quote standard calculations in deRham cohomology.

Raw LaTeX

Let \(\omega\) be a smooth closed \(p\)-form on \(S^n\) with \(p>0\). Show that there
is a smooth \((p-1)\)-form \(\alpha\) so that \(d\alpha=\omega\) if \(p<n\) but that this
need not be the case when \(p=n\). You are allowed to quote standard calculations in
deRham cohomology.

Topic Tags

Cohomology and Cup Products Differential Forms and Exterior Calculus Homology with Coefficients de Rham Cohomology