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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 1997 • Question 8

Question

Math 6510 • Differentiable Manifolds Fall 1997 • Question 8

Let \(M^n\subset \mathbb {R}^{n+1}\) be an embedded differentiable submanifold. Prove that for each \(P\in M\) there is a neighborhood \(U\subset \mathbb {R}^{n+1}\) of \(P\) so that \(U\cap M\) is expressible as the graph “above” at least one of the \(n+1\) coordinate hyperplanes of \(\mathbb {R}^{n+1}\).

Raw LaTeX

Let \(M^n\subset\mathbb{R}^{n+1}\) be an embedded differentiable
submanifold. Prove that for each \(P\in M\) there is a neighborhood
\(U\subset\mathbb{R}^{n+1}\) of \(P\) so that \(U\cap M\) is expressible as the graph
``above'' at least one of the \(n+1\) coordinate hyperplanes of \(\mathbb{R}^{n+1}\).

Topic Tags

Free Groups, Graph Coverings, and Presentations Smooth Manifolds and Embedded Submanifolds