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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 1998 • Question 5

Question

Math 6510 • Differentiable Manifolds Fall 1998 • Question 5

Let \(M^m,N^n\subset P^p\) be smooth embedded submanifolds of a smooth manifold. \(M^m\) and \(N^n\) are said to intersect transversally if \(T_xM+T_xN=T_xP\) for every \(x\in M\cap N\). Prove that if \(M^m\) and \(N^n\) intersect transversally then \(M^m\cap N^n\) is a smooth submanifold of \(P^p\). What is its dimension?

Raw LaTeX

Let \(M^m,N^n\subset P^p\) be smooth embedded submanifolds of a smooth manifold.
\(M^m\) and \(N^n\) are said to \emph{intersect transversally} if
\(T_xM+T_xN=T_xP\) for every \(x\in M\cap N\). Prove that if \(M^m\) and \(N^n\) intersect
transversally then \(M^m\cap N^n\) is a smooth submanifold of \(P^p\). What is its
dimension?

Topic Tags

Homotopy and Degree on Manifolds Smooth Manifolds and Embedded Submanifolds Stokes' Theorem and Integration on Manifolds Transversality and Intersection Theory