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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Winter 2005 • Question 4

Question

Math 6510 • Differentiable Manifolds Winter 2005 • Question 4

Let \(M\) be a manifold and \(\Delta \) a smooth \(k\)-plane field on \(M\). Suppose that \(X\) and \(Y\) are smooth vector fields on \(M\) with values in \(\Delta \). Show that if for some \(p\in M\) we have \(X_p=0\) then \([X,Y]_p\in \Delta \).

Raw LaTeX

Let \(M\) be a manifold and \(\Delta\) a smooth \(k\)-plane field on \(M\). Suppose that
\(X\) and \(Y\) are smooth vector fields on \(M\) with values in \(\Delta\). Show that if for
some \(p\in M\) we have \(X_p=0\) then \([X,Y]_p\in\Delta\).

Topic Tags

Distributions, Plane Fields, and Frobenius Integrability Left-Invariant Vector Fields Smooth Manifolds and Embedded Submanifolds Vector Fields and Flows