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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Spring 2023 • Question 3

Question

Math 6510 • Differentiable Manifolds Spring 2023 • Question 3

Let \(M\) and \(N\) be submanifolds of \(\mathbb {R}^n\) and for all \(a\in \mathbb {R}^n\) let \(M_a=\) \(\{x+a\in \mathbb {R}^n\mid x\in M\}\). Show that \(M_a\) and \(N\) are transverse for almost all choices of \(a\in \mathbb {R}^n\).

Raw LaTeX

Let \(M\) and \(N\) be submanifolds of \(\mathbb{R}^n\) and for all \(a\in\mathbb{R}^n\) let \(M_a=\)
\(\{x+a\in\mathbb{R}^n\mid x\in M\}\). Show that \(M_a\) and \(N\) are transverse for almost
all choices of \(a\in\mathbb{R}^n\).

Topic Tags

Smooth Manifolds and Embedded Submanifolds Transversality and Intersection Theory