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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2023 • Question 3

Question

Math 6510 • Differentiable Manifolds Fall 2023 • Question 3

Let \(R:S^2\to \mathbb {R}_+\) be a nonconstant \(C^\infty \) function. Show that \(F:S^2\to \mathbb {R}^3\) is defined by \(F(v)=R(v)v\), then there exists a value of \(r\in \mathbb {R}_+\) such that if \(S_r^2\) is the sphere of radius \(r\) centered at 0, then the intersection of \(S_r^2\) with the image of \(F\) is a nonempty finite union of circles.

Raw LaTeX

Let \(R:S^2\to\mathbb{R}_+\) be a nonconstant \(C^\infty\) function. Show that \(F:S^2\to\mathbb{R}^3\) is defined
by \(F(v)=R(v)v\), then there exists a value of \(r\in\mathbb{R}_+\) such that if \(S_r^2\) is the sphere of radius \(r\)
centered at 0, then the intersection of \(S_r^2\) with the image of \(F\) is a nonempty finite union of circles.

Topic Tags

Transversality and Intersection Theory