
\documentclass{amsart}

\usepackage{graphics,verbatim,color,amsthm}
\usepackage[all]{xy}
%package above ADDED to make nice commutative diagrams

\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{proposition}{Proposition}
\newtheorem{exer}{Exercise}
\newtheorem{lemma}{Lemma}  
\newtheorem{cor}{Corollary} 
\newtheorem{definition}{Definition}

\theoremstyle{remark}
\newtheorem*{remark}{Remark}

\theoremstyle{definition}
\newtheorem{example}{Example}


%{Exercise} 

 

%\newtheorem*{corollary}{Corollary}
 
\def\metric#1{\langle #1 \rangle}
 
\def\R{\mathbb{R}}
\def\Z{\mathbb{Z}}
\def\C{\mathbb{C}}
\def\K{\mathbb{K}}
\def\Q{\mathbb{Q}}
\def\CP{\mathbf{CP}}
\def\bP{\mathbb{P}}

\def\Gg{\mathfrak g}

 \newcommand{\ddt} 
{
{{d}   \over {d t}}|_{t=0} 
}
 
 \newcommand{\dd}[2]
{
{{\partial #1}   \over {\partial #2}}
}
%  a  macro for making partial derivates easily  

\begin{document}VECTOR FIELDS HOMEWORK assigments.

1.    Let $X$ be a vector field on a manifold, and p a point of the manifold
and suppose that $X(p) \ne 0$.  Prove that there is another vector field $Y$
on the manifold such that the equality $[X,Y] = X$ holds in a neighborhood of $p$.

\vskip .4cm

2.   Suppose that $X$ is a vector field on a manifold $M$ , and p is a point of the manifold. 
Use the coordinate definitions of vector fields (``An n-component vector  $X^1, \ldots , X^n$ depending
on the choice of coordinates which transforms under changes
of variables like...  '') to prove: 

(i) If $X(p) =0$ then the first derivative of $X$ at   $p$ is well-defined and can be viewed
as a linear map $DX_p:  T_p M \to T_p M$.  

(ii) If $X(p) \ne 0$ then the first derivative of $X$ at $p$, depends on the choice of coordinates
and hence is not well-defined.

\vskip .4cm


 3.   $Gl(2, \R)$ acts on $\R \bP^1$. Write the action as $(g, [x,y]) \mapsto g \cdot [x,y] = [a x + by, cx + dy ]$.
  By the standard affine chart let us mean the map $x \mapsto [x,1]$ or its inverse.
  
  A) Verify that this action is the standard linear fractional transformation upon being  viewed in the standard affine chart.
  
  B)   The associated infinitesimal action is the linear map $\sigma: gl(2, \R) \to \chi(\R \bP ^2)$
  which sends a   2 by 2 matrix $\xi \in gl(2, \R)$ to the  vector field $\sigma(\xi)$ on $\R \bP ^1$, the infinitesimal generator of
  this action: $\sigma (\xi) ([x,y]) = \ddt exp(t \xi) \cdot [x,y]$.  
  Write out   the infinitesimal generator map {\it explicitly} by writing out, rel. to the standard affine coordinate,  the   four vector fields
  corresponding to the image of the standard basis $E_1, E_2, E_3, E_4$  of $gl(2, \R)$ under $\sigma$.  Your answers will be of the form
  $\sigma(E_i) = f_i (x) \dd{}{x}$, with explicit functions $f_i (x)$,  $i = 1,2, 3, 4$.
  
   
 




\end{document}

