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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Winter 2007 • Question 5

Question

Math 6510 • Differentiable Manifolds Winter 2007 • Question 5

Let \(f:\mathbb {R}^n\to \mathbb {R}\) be a smooth function. Show that for every \(\epsilon >0\) there is \(v\in \mathbb {R}^n\) with \(|v|<\epsilon \) such that the function \(g_v:\mathbb {R}^n\to \mathbb {R}\) given by \[ g_v(x)=f(x)+v\cdot x \] is Morse.

Raw LaTeX

Let \(f:\mathbb{R}^n\to\mathbb{R}\) be a smooth function. Show that for every
\(\epsilon>0\) there is \(v\in\mathbb{R}^n\) with \(|v|<\epsilon\) such that the function
\(g_v:\mathbb{R}^n\to\mathbb{R}\) given by
\[
  g_v(x)=f(x)+v\cdot x
\]
is Morse.

Topic Tags

Morse Theory and Critical Points