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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Spring 2019 • Question 5

Question

Math 6510 • Differentiable Manifolds Spring 2019 • Question 5

Let \(M\) be a compact, smooth two dimensional manifold without boundary embedded in \(\mathbb {R}^3\). Show that for almost every two-plane \(L\in \mathbb {R}^3\) such that \(L\cap M\) is nonempty, the intersection \(L\cap M\) consists of finitely many closed curves.

Raw LaTeX

Let \(M\) be a compact, smooth two dimensional manifold without boundary
embedded in \(\R^3\). Show that for almost every two-plane \(L\in\R^3\) such that
\(L\cap M\) is nonempty, the intersection \(L\cap M\) consists of finitely many
closed curves.

Topic Tags

Homotopy and Degree on Manifolds Smooth Manifolds and Embedded Submanifolds Stokes' Theorem and Integration on Manifolds Transversality and Intersection Theory