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Question

Math 6510 • Differentiable Manifolds

Spring 2015 • Question 1

Question

Math 6510 • Differentiable Manifolds Spring 2015 • Question 1

Let \(S^2=\{(x,y,z)\in \mathbb {R}^3\mid x^2+y^2+z^2=1\}\) be the two sphere in \(\mathbb {R}^3\). Define \(f:\mathbb {R}^3\to \mathbb {R}\) by \(f(x,y,z)=x(y-2)\).

  1. Show that 0 is a critical value of \(f\) and find all critical points in \(\mathbb {R}^3\).
  2. Show that 0 is a regular value of \(f\) restricted to \(S^2\) and conclude that \(M=f^{-1}(0)\cap S^2\) is manifold. What manifold is \(M\)?
  3. Define \(g:\mathbb {R}^3\to \mathbb {R}\) by \(g(x,y,z)=x+z\). Calculate the critical points of \(g\) restricted to \(S^2\) and the critical points of \(g\) restricted to \(M\).

Raw LaTeX

Let \(S^2=\{(x,y,z)\in\R^3\mid x^2+y^2+z^2=1\}\) be the two sphere in
\(\R^3\). Define \(f:\R^3\to\R\) by \(f(x,y,z)=x(y-2)\).
\begin{enumerate}
\item[(a)] Show that 0 is a critical value of \(f\) and find all critical points in
\(\R^3\).
\item[(b)] Show that 0 is a regular value of \(f\) restricted to \(S^2\) and
conclude that \(M=f^{-1}(0)\cap S^2\) is manifold. What manifold is \(M\)?
\item[(c)] Define \(g:\R^3\to\R\) by \(g(x,y,z)=x+z\). Calculate the critical points
of \(g\) restricted to \(S^2\) and the critical points of \(g\) restricted to \(M\).
\end{enumerate}

Topic Tags

Homotopy and Degree on Manifolds Morse Theory and Critical Points Regular Values and Level Sets Smooth Manifolds and Embedded Submanifolds