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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Spring 2023 • Question 1

Question

Math 6510 • Differentiable Manifolds Spring 2023 • Question 1

Define \(F:\mathbb {R}^3\to \mathbb {R}^2\) by \(F(x,y,z)=(x^2+y^2-2,z)\) and let \(S^1=\) \(\{(x,y)\in \mathbb {R}^2\mid x^2+y^2=1\}\) be the unit circle. Show that \(F^{-1}(S^1)\) is a smooth manifold and calculate its dimension.

Raw LaTeX

Define \(F:\mathbb{R}^3\to\mathbb{R}^2\) by \(F(x,y,z)=(x^2+y^2-2,z)\) and let \(S^1=\)
\(\{(x,y)\in\mathbb{R}^2\mid x^2+y^2=1\}\) be the unit circle. Show that \(F^{-1}(S^1)\) is a
smooth manifold and calculate its dimension.

Topic Tags

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