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  \begin{document}


  
  \vskip 1cm

  \centerline{\Large {Tangent Spaces and Morse functions}}
\vskip 1cm
  


\subsection*{Tangent spaces}

\begin{enumerate}[1.]

  \item Show that for $k$ odd the $k$-sphere $S^k$ admits a nowhere
    vanishing vector field. Note: It is a deep result that the only
    spheres with a {\it trivial} tangent bundle are $S^1,S^3$ and
    $S^7$.

  \item Let $G$ be a Lie group and $v\in T_1G$ a vector in the tangent
    space of the identity. For $g\in G$ define the vector $V(g)\in
    T_g(G)$ as
    $$V(g)=dL_g(v)$$
    where $L_g:G\to G$ is the diffeomorphism defined by $x\mapsto gx$
    (left translation by $g$). Show that $V$ is a smooth vector field
    on $G$. Deduce that Lie groups have a trivial tangent
    bundle. Hint: If $m:G\times G\to G$ is multiplication, show that
    the following composition is $V$:
    $$G\to T(G)\times T(G)\to T(G)$$ where the first map is inclusion
    $g\mapsto ((g,0),(1,v))$ (i.e. zero section in the first
    coordinate and constant $v\in T_1(G)$ in the second), and the
    second map is derivative $dm$ of $m$.

  

  

\end{enumerate}

\subsection*{Morse functions}

  
  
\begin{enumerate}[resume]

    \item Analyze the behavior of the given function $f:\R^2\to\R$ at
      the origin. Is the critical point nondegenerate? If it is,
      determine the index.
      \begin{enumerate}[(a)]
      \item $f(x,y)=x^2+4y^3$
      \item $f(x,y)=x^2-2xy+y^2$
      \item $f(x,y)=x^2+y^4$
      \item $f(x,y)=x^2+11xy+y^2/2+x^6$
      \item $f(x,y)=10xy+y^2+75y^3$
      \end{enumerate}

      \item Let $S^n=\{(x_1,x_2,\cdots,x_{n+1})\in\R^{n+1}\mid \sum
        x_i^2=1\}$ and define $f:S^n\to\R$ by
        $f(x_1,x_2,\cdots,x_{n+1})=x_{n+1}$. Show that $f$ is a Morse
        function, and compute all critical points and their indices.

        \item View $\R P^2$ as the quotient of the antipodal map
          $a:S^2\to S^2$, $a(x,y,z)=(-x,-y,-z)$, where $S^2\subset
          \R^3$ is the unit sphere as usual. Show that the map $f:\R
          P^2\to\R$ defined by $f([x,y,z])=x^2+2y^2+3z^2$ is a Morse
          function and compute all critical points and their indices.

        \item \label{6}
          Let $U\subseteq \R^n$ be open. Show that a smooth
            function $f:U\to\R$ is Morse iff $$\det(H)^2+\sum_i \bigg(\frac
            {\partial f}{\partial x_i}\bigg)^2>0$$ where
            $H=\bigg(\frac {\partial ^2 f}{\partial x_i\partial x_j}\bigg)$
            is the Hessian.

            \item Prove the stability of Morse functions: Let $X$ be a
              compact manifold and $F:X\times P\to\R$ smooth for a
              manifold $P$. If $F_{p_0}$ is Morse for $p_0\in P$, show
              that $F_p$ is Morse for all $p$ in a neighborhood of
              $p_0$. Hint: Use Problem \ref{6}.

              \item Let $X$ be a compact manifold. Show that there is
                a Morse function $f:X\to\R$ that has distinct values
                at distinct critical points of $X$. Hint: Start with a
                Morse function $g:X\to\R$ with critical points
                $x_1,\cdots,x_k$, fix mesa functions $\rho_i:X\to
                [0,1]$ that are 1 in a small neighborhood of $x_i$ and
                0 outside a slightly larger neighborhood. Then let
                $f=g+\sum a_i\rho_i$ for small and generic $a_i$.


\end{enumerate}
      
\end{document}
