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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2016 • Question 5

Question

Math 6510 • Differentiable Manifolds Fall 2016 • Question 5

Suppose \(f:M\to N\) is a smooth function of smooth manifolds. Suppose \(Q\) is an embedded submanifold of \(N\), and that \(f\) is transverse to \(Q\). Prove that if \(p\in f^{-1}(Q)\), then \[ T_p(f^{-1}(Q))=(D_pf)^{-1}[T_{f(p)}Q]. \] (The above line says that the tangent space for \(f^{-1}(Q)\) at \(p\) is the inverse image of the tangent space of \(Q\) at \(f(p)\) under the differential of \(f\) at \(p\).)

Raw LaTeX

Suppose \(f:M\to N\) is a smooth function of smooth manifolds. Suppose \(Q\) is
an embedded submanifold of \(N\), and that \(f\) is transverse to \(Q\). Prove that if
\(p\in f^{-1}(Q)\), then
\[
  T_p(f^{-1}(Q))=(D_pf)^{-1}[T_{f(p)}Q].
\]
(The above line says that the tangent space for \(f^{-1}(Q)\) at \(p\) is the inverse
image of the tangent space of \(Q\) at \(f(p)\) under the differential of \(f\) at \(p\).)

Topic Tags

Differential Forms and Exterior Calculus Smooth Manifolds and Embedded Submanifolds Tangent Spaces and Differential Maps Transversality and Intersection Theory