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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2017 • Question 3

Question

Math 6510 • Differentiable Manifolds Fall 2017 • Question 3

Let \(M\) be the smooth submanifold of \(\mathbb {R}^3\) that is the solution set of the equation \(x^2+y^2-z^2=1\). Let \(N\) be the smooth submanifold of \(\mathbb {R}^3\) that is the solution set of the equation \(x^2+y^2+z^2=1\). Prove that \(M\) and \(N\) do not intersect transversally in \(\mathbb {R}^3\).

Raw LaTeX

Let \(M\) be the smooth submanifold of \(\R^3\) that is the solution set of the
equation \(x^2+y^2-z^2=1\). Let \(N\) be the smooth submanifold of \(\R^3\) that is the
solution set of the equation \(x^2+y^2+z^2=1\). Prove that \(M\) and \(N\) do not
intersect transversally in \(\R^3\).

Topic Tags

Smooth Manifolds and Embedded Submanifolds Transversality and Intersection Theory