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% Common mathematical symbols
\newcommand{\RR}{\mathbb{R}}
\newcommand{\CC}{\mathbb{C}}
\newcommand{\ZZ}{\mathbb{Z}}

\begin{document}
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\lhead{1300F Geometry and Topology, Assignment 2}
\rhead{Due date: October 23, 2017}
\cfoot{}


\begin{exercise}
  Construct, using the stereographic charts for $S^2$ given in class,
  a smooth vector field on $S^2$ which vanishes exactly at 2 points,
  and another vector field which vanishes at exactly 1 point.
\end{exercise}

\begin{exercise}
Determine whether or not the vector field $(1 + x^2)\frac{\partial}{\partial x}$ is complete on $\RR$.
\end{exercise}

\begin{exercise}
Let $M=\RR^n$, with coordinates $(x^1,\ldots, x^n)$. Then $TM = \RR^n \times \RR^n$.
\begin{enumerate}
\item The trivial map $E:M\to TM$ given by $x\mapsto (x,x)$ defines a section of the
  tangent bundle, i.e. a vector field.  Write the vector field in the given coordinates.  Compute the time-$t$ flow of
  this vector field and determine whether it is complete or not. Draw a picture of the vector field in the cases $n=1$ and $n=2$.
\item Suppose $A:\RR^n\to \RR^n$ is a linear map with matrix coefficients $a_{ij}$. Then the map $A:x\mapsto
  (x,Ax)$ defines a vector field on $M$;  Write the vector field in coordinates, compute its flow and
  determine if it is complete.
\end{enumerate}
\end{exercise}

\begin{exercise}
Prove that the orthogonal group $O(n,\RR)=\{X\in GL(n,\RR) :  XX^\top = 1\}$ is a smooth submanifold of $M(n,\RR)$, the $n\times n$ matrices.  To show this, consider the map $f:M(n,\RR)\to S(n,\RR)$ to the symmetric matrices $S(n,\RR)$ given by $f(X) = XX^\top$.  
\end{exercise}

\begin{exercise}\mbox{}
  Prove that if $K$ is a submanifold of $L$ and $L$ is a submanifold of $M$, then $K$ is
  a submanifold of $M$.
\end{exercise}


\begin{exercise}\mbox{}
\begin{enumerate}
\item
Let $\varphi:\RR P^2\to\RR^3$ be the map defined by 
\[
\varphi([x:y:z]) = \frac{1}{x^2+y^2+z^2} (yz, xz, xy).
\]
Show that $\varphi$ is smooth.  Is it an immersion? If not, at which points in the domain does $\varphi$ fail to be an immersion? 


\item
Let $\Phi:\RR P^2\to \RR^4$ be the map 
\[
\Phi([x:y:z]) = \frac{1}{x^2+y^2+z^2} (x^2-z^2,yz, xz, xy).
\]
Show that this is an embedding of $\RR P^2$ into $\RR^4$.
\end{enumerate}
\end{exercise}





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