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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2012 • Question 9

Question

Math 6510 • Differentiable Manifolds Fall 2012 • Question 9

Let \(M\) be a compact, connected smooth manifold, and suppose that \(N\) is a connected smooth manifold. If \(F:M\times [0,1]\to N\) is smooth and \(m\mapsto F(m,0)\) is an immersion of \(M\) into \(N\), then show there is some \(\varepsilon >0\) such that \(m\mapsto F(m,\delta )\) is an immersion of \(M\) into \(N\) for any fixed \(\delta <\varepsilon \).

Raw LaTeX

Let \(M\) be a compact, connected smooth manifold, and suppose that \(N\) is a
connected smooth manifold. If \(F:M\times[0,1]\to N\) is smooth and
\(m\mapsto F(m,0)\) is an immersion of \(M\) into \(N\), then show there is some
\(\varepsilon>0\) such that \(m\mapsto F(m,\delta)\) is an immersion of \(M\) into \(N\)
for any fixed \(\delta<\varepsilon\).

Topic Tags

Homotopy and Degree on Manifolds Immersions, Submersions, and Embeddings Smooth Manifolds and Embedded Submanifolds Stokes' Theorem and Integration on Manifolds