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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2017 • Question 4

Question

Math 6510 • Differentiable Manifolds Fall 2017 • Question 4

Let \(M\) and \(N\) be smooth manifolds, and let \(M\) be compact. Suppose \(F:M\times [0,1]\to N\) is a smooth function, and for any \(t\in [0,1]\) let \(F_t:M\to N\) be the function defined by \(F_t(p)=F(p,t)\). If \(F_0\) is an immersion, prove that there is some \(\varepsilon \in (0,1]\) such that \(F_t\) is an immersion for all \(t\in [0,\varepsilon )\).

Raw LaTeX

Let \(M\) and \(N\) be smooth manifolds, and let \(M\) be compact. Suppose
\(F:M\times[0,1]\to N\) is a smooth function, and for any \(t\in[0,1]\) let
\(F_t:M\to N\) be the function defined by \(F_t(p)=F(p,t)\). If \(F_0\) is an immersion,
prove that there is some \(\varepsilon\in(0,1]\) such that \(F_t\) is an immersion for
all \(t\in[0,\varepsilon)\).

Topic Tags

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