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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2022 • Question 1

Question

Math 6510 • Differentiable Manifolds Fall 2022 • Question 1

Let \(S^2=\{(x,y,z)\in \mathbb {R}^3\mid x^2+y^2+z^2=1\}\) be the 2-sphere and let \(M_a=f_a^{-1}(0)\) where \(a\in \mathbb {R}\) and \(f_a:\mathbb {R}^3\to \mathbb {R}\) is the function \(f_a(x,y,z)=x^2+y^2-z+a\). For what values of \(a\in \mathbb {R}\) are \(S^2\) and \(M_a\) transverse? Justify your answer.

Raw LaTeX

Let \(S^2=\{(x,y,z)\in\mathbb{R}^3\mid x^2+y^2+z^2=1\}\) be the 2-sphere and let
\(M_a=f_a^{-1}(0)\) where \(a\in\mathbb{R}\) and \(f_a:\mathbb{R}^3\to\mathbb{R}\) is the function
\(f_a(x,y,z)=x^2+y^2-z+a\). For what values of \(a\in\mathbb{R}\) are \(S^2\) and \(M_a\) transverse?
Justify your answer.

Topic Tags

Transversality and Intersection Theory