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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2007 • Question 3

Question

Math 6510 • Differentiable Manifolds Fall 2007 • Question 3

Let \(\omega \) be an \(n\)-form on \(\mathbb {R}^n\) with compact support. Show that there exists a compactly supported \((n-1)\) form \(\eta \) with \(\omega =d\eta \) if and only if \[ \int _{\mathbb {R}^n}\omega =0. \] You are allowed to use the standard results about deRham cohomology of \(\mathbb {R}^n\) and spheres.

Raw LaTeX

Let \(\omega\) be an \(n\)-form on \(\R^n\) with compact support. Show that there
exists a compactly supported \((n-1)\) form \(\eta\) with \(\omega=d\eta\) if and only if
\[
  \int_{\R^n}\omega=0.
\]
You are allowed to use the standard results about deRham cohomology of \(\R^n\) and
spheres.

Topic Tags

Cohomology and Cup Products Compactly Supported Forms and Integration Differential Forms and Exterior Calculus de Rham Cohomology