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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 1994 • Question 10

Question

Math 6510 • Differentiable Manifolds Fall 1994 • Question 10

Let \(f\in C^\infty (S^n)\) be a smooth function and let \(x_1,x_2\in S^n\) be two points such that \(f(x_1)<0<f(x_2)\). Let \(\omega \) be a smooth nonvanishing \(n\)-form. Then there is a diffeomorphism \(\varphi :S^n\to S^n\) such that \[ \int _{S^n}(\varphi ^*f)\,\omega =0. \]

II. Ten point questions.

Raw LaTeX

Let \(f\in C^\infty(S^n)\) be a smooth function and let
\(x_1,x_2\in S^n\) be two points such that \(f(x_1)<0<f(x_2)\). Let \(\omega\) be a smooth
nonvanishing \(n\)-form. Then there is a diffeomorphism \(\varphi:S^n\to S^n\) such that
\[
  \int_{S^n}(\varphi^*f)\,\omega=0.
\]

\textbf{II. Ten point questions.}

Topic Tags

Diffeomorphisms and Isotopy Methods Differential Forms and Exterior Calculus Fixed Point Theorems and Borsuk-Ulam