Math 6510 • Differentiable Manifolds

Let \(S^2=\{(x,y,z)\in\R^3:x^2+y^2+z^2=1\}\) be the two sphere in three
space. Consider the vector fields
\[
  V=y\frac{\partial}{\partial z}-z\frac{\partial}{\partial y},
  \qquad
  W=z\frac{\partial}{\partial x}-x\frac{\partial}{\partial z}.
\]
\begin{enumerate}
\item Show \(V\) and \(W\) are tangent to \(S^2\), and hence by restriction define
vector fields on \(S^2\).

\item Find the flow \(\Theta_t^V(x,y,z)\) on \(S^2\) generated by \(V\).

\item Compute \((\Theta_h^V)_*W\) on \(S^2\).

\item Compute the Lie Derivative \(\mathcal{L}_VW\) on \(S^2\) in two ways.
\end{enumerate}