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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2004 • Question 2

Question

Math 6510 • Differentiable Manifolds Fall 2004 • Question 2

Regard the real projective plane \(\mathbb {R}P^2\) as the space of triples \((x,y,z)\in \mathbb {R}^3\setminus \{0\}\) modulo the relation \((x,y,z)\sim (tx,ty,tz)\) for \(t\in \mathbb {R}\setminus \{0\}\). Define \[ f:\mathbb {R}P^2\to \mathbb {R} \] by \[ f(x,y,z)=\frac {x^2+2y^2}{x^2+y^2+z^2}. \] Compute all critical points of \(f\). Show that \(f\) is a Morse function and compute the Morse index of each critical point. (Hint: Work in local coordinates, in standard charts \(U_0,U_1,U_2\) where \(U_0=\{x\ne 0\}\) etc.)

Raw LaTeX

Regard the real projective plane \(\mathbb{R}P^2\) as the space of triples
\((x,y,z)\in\mathbb{R}^3\setminus\{0\}\) modulo the relation
\((x,y,z)\sim(tx,ty,tz)\) for \(t\in\mathbb{R}\setminus\{0\}\). Define
\[
  f:\mathbb{R}P^2\to\mathbb{R}
\]
by
\[
  f(x,y,z)=\frac{x^2+2y^2}{x^2+y^2+z^2}.
\]
Compute all critical points of \(f\). Show that \(f\) is a Morse function and compute the
Morse index of each critical point. (Hint: Work in local coordinates, in standard
charts \(U_0,U_1,U_2\) where \(U_0=\{x\ne0\}\) etc.)

Topic Tags

Morse Theory and Critical Points Real Projective Spaces Surface Topology and Surface Groups