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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2021 • Question 2

Question

Math 6510 • Differentiable Manifolds Fall 2021 • Question 2

Let \(N=\{(x,y,z,w)\in \mathbb {R}^4:x^2+y^2=z^2+w^2=1\}\) and \(f(p,q,r):\) \(\mathbb {R}^3\to \mathbb {R}^4\) be given by \(x=p+r\), \(y=q-r\), \(z=p-r\), \(w=p+q\). Show that \(f^{-1}(N)\) an embedded submanifold of \(\mathbb {R}^3\). What is its dimension?

Raw LaTeX

Let \(N=\{(x,y,z,w)\in\mathbb{R}^4:x^2+y^2=z^2+w^2=1\}\) and \(f(p,q,r):\)
\(\mathbb{R}^3\to\mathbb{R}^4\) be given by \(x=p+r\), \(y=q-r\), \(z=p-r\), \(w=p+q\). Show
that \(f^{-1}(N)\) an embedded submanifold of \(\mathbb{R}^3\). What is its dimension?

Topic Tags

Smooth Manifolds and Embedded Submanifolds