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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Spring 2009 • Question 2

Question

Math 6510 • Differentiable Manifolds Spring 2009 • Question 2

Let \(U\) be the open set in \(\mathbb {R}^3\), \(U=\{(x,y,z):x>0,\ y>0,\ z>0\}\) and let \(D\) be the distribution in \(U\) spanned by the vector fields: \[ V_1=y\frac {\partial }{\partial z}-z\frac {\partial }{\partial y} \qquad V_2=z\frac {\partial }{\partial x}-x\frac {\partial }{\partial z} \] Show that \(D\) is involutive. What does the Frobenius Theorem imply about \(D\)?

Raw LaTeX

Let \(U\) be the open set in \(\R^3\),
\(U=\{(x,y,z):x>0,\ y>0,\ z>0\}\) and let \(D\) be the distribution in \(U\) spanned
by the vector fields:
\[
  V_1=y\frac{\partial}{\partial z}-z\frac{\partial}{\partial y}
  \qquad
  V_2=z\frac{\partial}{\partial x}-x\frac{\partial}{\partial z}
\]
Show that \(D\) is involutive. What does the Frobenius Theorem imply about \(D\)?

Topic Tags

Distributions, Plane Fields, and Frobenius Integrability Left-Invariant Vector Fields Vector Fields and Flows