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\title{21-236: Analysis Homework 5}
\author{David Handron}

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\maketitle


    Submit the \LaTeX\ file containing your solutions to the
    email address analysishw@legba.math.cmu.edu by
    11:59pm on Wednesday 16 February.   

 


\begin{enumerate}

\item \label{bundle} Let $U$ be an open subset of $\mathbb{R}^n$.
  Give a reasonable definition of the tangent bundle of $U$, $TU$.
  That is to say, identify the base space $B$, total space $E$ and
  projection $\pi:E\to B$, as well as the vector space operations on
  each fibre, $T_aU=\pi^{-1}(a)$.

\vskip.15in

\item Show, using your definition from Problem \ref{bundle}, that
  there is a bundle equivalence (as defined in class) $TU\cong TM|_U$,
  i.e.
$$
TU\cong \cup_{a\in U} T_a\mathbb{R}^n.
$$

\item Suppose $U$ is open in $\mathbb{R}^n$ and $W$ is open in
$\mathbb{R}^m$.  Let $f:U\to W$ be a smooth map.

\begin{enumerate}

\item Let $X$ be a vector field on U (i.e. a section of $TU$).  Show
  that in general, it is not possible to define a unique vector field
  $Y$ on $W$, such that $Df_a(X_a)=Y_{f(a)}$. [Note: there are two
  potential problems, you should identify them both.] {\em Thus,
  although individual tangent vectors ``push forward'' under Df,
  vector fields do not.}

\item What conditions can be placed on
  the map $f$ that would allow a vector field on $U$ to be pushed
  forward to $W$?


\item Suppose that $m=n$ and $\det(f'(a))\neq 0$ for all $a\in U$.
  Let $Y$ be a vector field on $W$.  Show that there exists a unique
  vector field $X$ on $U$ such that $Df_a(X_a)=Y_{f(a)}$.  That is,
  there exists a section $X$ of $TU$ such that the diagram
$$
\begin{CD}
TU @>{f_*}>> TW \\
@A{X}AA @AA{Y}A \\
U @>{f}>> W
\end{CD}
$$ 
commutes.  {\em Thus, in certain situations, vector fields can be
``pulled back'' but not ``pushed forward''.}

\end{enumerate}


\end{enumerate}

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