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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2009 • Question 6

Question

Math 6510 • Differentiable Manifolds Fall 2009 • Question 6

Let \(S^n=\{v\in \mathbb {R}^{n+1}\mid \lVert v\rVert =1\}\) and let \(f:S^n\to S^n\) be the restriction of the map \(\mathbb {R}^{n+1}\to \mathbb {R}^{n+1}\) given by \(v\mapsto -v\). Suppose \(\omega \) is an \(n\)-form on \(S^n\) with \(\int _{S^n}\omega =\frac {\pi ^2}{6}\). What is \(\int _{S^n}f^*\omega \). Why?

Raw LaTeX

Let \(S^n=\{v\in\R^{n+1}\mid \lVert v\rVert=1\}\) and let
\(f:S^n\to S^n\) be the restriction of the map \(\R^{n+1}\to\R^{n+1}\) given by
\(v\mapsto -v\). Suppose \(\omega\) is an \(n\)-form on \(S^n\) with
\(\int_{S^n}\omega=\frac{\pi^2}{6}\). What is \(\int_{S^n}f^*\omega\). Why?

Topic Tags

Differential Forms and Exterior Calculus Fixed Point Theorems and Borsuk-Ulam