Math 6510 • Differentiable Manifolds

Let \(V\) and \(W\) be vector fields on \(\R^3\) given by
\[
  V(x,y,z)=\frac{\partial}{\partial x}
  -y\frac{\partial}{\partial z}
  \quad\text{and}\quad
  W(x,y,z)=\frac{\partial}{\partial y}
  +x^2\frac{\partial}{\partial z}.
\]
\begin{enumerate}
\item[(a)] Find \([V,W](x,y,z)\).
\item[(b)] What is the definition of an integrable, \(k\)-dimensional plane field on
a manifold \(M\)?
\item[(c)] Prove or disprove:
\(\Delta(x,y,z)=\spanop\{V(x,y,z),W(x,y,z)\}\) is an integrable plane field.
\item[(d)] Give an example of three foliations on a two dimensional torus: one with
every leaf compact; one with every leaf noncompact; and one with some compact and
some noncompact leaves.
\end{enumerate}