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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Spring 2014 • Question 5

Question

Math 6510 • Differentiable Manifolds Spring 2014 • Question 5

Let \(S^1=\{(x,y):x^2+y^2=1\}\) and define \(f:\mathbb {R}\to S^1\) by \(f(t)=(\cos 2\pi t,\sin 2\pi t)\). Show there exists a unique 1-form \(\omega \) on \(S^1\) such that \(f^*\omega =dt\). Show that \(\omega \) is closed but not exact.

Raw LaTeX

Let \(S^1=\{(x,y):x^2+y^2=1\}\) and define \(f:\R\to S^1\) by
\(f(t)=(\cos2\pi t,\sin2\pi t)\). Show there exists a unique 1-form \(\omega\) on
\(S^1\) such that \(f^*\omega=dt\). Show that \(\omega\) is closed but not exact.

Topic Tags

Differential Forms and Exterior Calculus de Rham Cohomology