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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Spring 2021 • Question 2

Question

Math 6510 • Differentiable Manifolds Spring 2021 • Question 2

Let \(M\subset \mathbb {R}^3\) be a submanifold diffeomorphic to the circle. Prove that for every \(\epsilon > 0\) there is a vector \(v\in \mathbb {R}^3\) of norm \(<\epsilon \) such that \(M\) and \(M+v=\{x+v\mid x\in M\}\) are disjoint. Carefully state any theorems you are citing.

Raw LaTeX

Let \(M\subset\mathbb{R}^3\) be a submanifold diffeomorphic to the circle. Prove that
for every \(\epsilon > 0\) there is a vector \(v\in\mathbb{R}^3\) of norm \(<\epsilon\) such that \(M\) and
\(M+v=\{x+v\mid x\in M\}\) are disjoint. Carefully state any theorems
you are citing.

Topic Tags

Smooth Manifolds and Embedded Submanifolds