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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 1999 • Question 7

Question

Math 6510 • Differentiable Manifolds Fall 1999 • Question 7

Let \(\Sigma ^2\) be a (nonempty) smooth closed (compact without boundary) surface smoothly embedded in \(\mathbb {R}^3\). Show that there is a two-plane \(P\) in \(\mathbb {R}^3\) which intersects \(\Sigma \) such that \(P\cap \Sigma \) consists of finitely many disjoint smooth closed loops.

Raw LaTeX

Let \(\Sigma^2\) be a (nonempty) smooth closed (compact without boundary) surface
smoothly embedded in \(\mathbb{R}^3\). Show that there is a two-plane \(P\) in
\(\mathbb{R}^3\) which intersects \(\Sigma\) such that \(P\cap\Sigma\) consists of finitely
many disjoint smooth closed loops.

Topic Tags

Smooth Manifolds and Embedded Submanifolds Surface Topology and Surface Groups