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% Common mathematical symbols
\newcommand{\RR}{\mathbb{R}}
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\begin{document}
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\lhead{1300F Geometry and Topology, Assignment 4}
\rhead{Due date: November 23, 2012}
\cfoot{}


\begin{exercise}
  Let $f:M\to\RR$ be a proper submersion.  Then $V = \ker Tf$ defines
  a codimension 1 subbundle of $TM$ called the vertical bundle.
\begin{enumerate}
\item Show, using a partition of unity, that it is possible to choose
  a rank 1 subbundle $H\subset TM$ complementary to $V$.  Do not use a
  Riemannian metric.
\item Conclude that to any vector field $v$ on $\RR$ we may associate
  a unique vector field $v^{h}$ on $M$ which lies in $H$.  This is
  called the horizontal lift of $v$.
\item Prove the preimages of any pair of points in the image of $f$
  are diffeomorphic manifolds.
\end{enumerate}
\end{exercise}


\begin{exercise}
  Consider the pair of vector fields $V = \partial_y$ and $W =
  y\partial_x - \partial_z$ on $\RR^3$, where we use coordinates
  $(x,y,z)$.  Is it possible to find a 2-dimensional submanifold of
  $\RR^3$ with the property that both $V$ and $W$ are tangent to it at
  all its points? If so, construct one; if not, why not?
  \end{exercise}


\begin{exercise}
  Let $\alpha = dz - xdy$ and $\beta = dx - wdy$ be 1-forms on $\RR^4$
  (or you can think of them as functions on $T\RR^4$).  Both $\alpha$ and $\beta$ have 
3-dimensional kernel on each tangent space.  
\begin{enumerate}
\item Prove that $\ker \alpha \cap \ker \beta$ has dimension 2.
\item Give a local basis $(V,W)$ for the above intersection.
\item Compute $[V,W]$. Is it linearly dependent on $(V,W)$?
\item Is it possible to find a 3-dimensional submanifold such that
  $V$, $W$, and $[V,W]$ are everwhere tangent to it? If so, construct one; if not, show why not.
\end{enumerate}
\end{exercise}


\begin{exercise}
Let $V$ be a finite dimensional vector space, and view it as a manifold $M$. Then $TM = V\times V$.
\begin{enumerate}
\item The trivial map $E:x\mapsto (x,x)$ defines a section of the
  tangent bundle, i.e. a vector field.  Compute the time-$t$ flow of
  this vector field and determine whether it is complete or not.
\item Suppose $A:V\to V$ is a linear map. Then the map $A:x\mapsto
  (x,Ax)$ defines a vector field on $M$; compute its flow and
  determine if it is complete.
\item If $A,B$ are linear maps as above, compute the Lie derivative of
  the vector fields they determine.  Verify the fact that if the
  vector fields commute, then the flows commute.
\end{exercise}

\end{document}
