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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Spring 2019 • Question 1

Question

Math 6510 • Differentiable Manifolds Spring 2019 • Question 1

Let \(M\) and \(N\) be smooth manifolds and let \(M\) be compact. Let \(F:M\times \mathbb {R}\to N\) be a smooth function and for any \(t\in \mathbb {R}\), let \(F_t:M\to N\) be the function defined by \(F_t(x)=F(x,t)\). If \(F_0\) is a submersion, prove that there is some \(\epsilon >0\) such that \(F_t\) is an submersion for all \(t\in (-\epsilon ,\epsilon )\).

Raw LaTeX

Let \(M\) and \(N\) be smooth manifolds and let \(M\) be compact. Let
\(F:M\times\R\to N\) be a smooth function and for any \(t\in\R\), let
\(F_t:M\to N\) be the function defined by \(F_t(x)=F(x,t)\). If \(F_0\) is a
submersion, prove that there is some \(\epsilon>0\) such that \(F_t\) is an
submersion for all \(t\in(-\epsilon,\epsilon)\).

Topic Tags

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