\magnification 2000
\parskip 10pt
\parindent 0pt
\hoffset -0.3truein
\hsize 7truein

\voffset -0.3truein
\vsize 9.8truein

\def\emph{}

\def\S{\Sigma_{i=1}^n}
\def\Sm{\Sigma_{i=1}^m}

\def\s{{\sigma}}
\def\ep{\epsilon}
\def\f{\vskip 10pt}
\def\h{\hskip 10pt}
\def\u{\vskip -8pt}
\def\bh{\hfil \break}

\def\N{{\bf N}}
\def\S{\Sigma_{i=1}^n}

\def\0{{\bf 0}}
\def\Z{{\bf Z}}
\def\R{{\bf R}}
\def\C{{\bf C}}
\def\x{{\bf x}}
\def\a{{\bf a}}
\def\y{{\bf y}}
\def\e{{\bf e}}
\def\i{{\bf i}}
\def\j{{\bf j}}
\def\k{{\bf k}}
\def\p{\psi}
\def\f{\phi}
\def\s{\sigma}
\def\ep{\epsilon}
\def\f{\vskip 10pt}
\def\u{ \vskip -5pt }
\def\h{\vskip -5pt \hskip 20pt}



$TM= \cup_{p \in M} T_p(M) = \left\{  \left(  p,v\right)  \mid p\in M,v\in T_{p}M\right\}  $, 

let $\pi$:$~TM\rightarrow M$ be defined by $\pi(p,v)=p$. 

Let $(\phi, U)$ be a chart for $M$.

If $q \in U$, let $\{ ( {\partial \over \partial x_1} )_q, ..., ( {\partial \over \partial x_m} )_q  \}$ be a basis (w.r.t $(\phi, U)$)
for $T_q(M) = T_q$


$t_{\phi}$:$~\pi^{-1}(U) \rightarrow \phi(U)\times{\R}^{m} \subset \R^{2m}$,

$t_{\phi}(q, v) = (\phi(q), a_1, ..., a_m)$ where $v = \Sm a_i ( {\partial \over \partial x_i} )_q$


Let ${\cal A}$ be a maximal atlas for $M$.

Basis for topology on $TM:$  \bh
$\{W ~|~ \exists (\phi, U) \in {\cal A}$  s.t. $W \subset \pi^{-1}(U)$ and 
$ t_\phi(W)$ open in $\R^{2m} \}$

Claim:  $TM$ is a $2m-$manifold and \bh
${\cal C} = \{(t_\phi, \pi^{-1}(U)) ~|~ (\phi, U) \in {\cal A}\}$ is a 
pre-atlas for $TM$. 

 $\pi$:$~TM\rightarrow M$, $\pi(p,v)=p$ is smooth 

$df:  TM \rightarrow TN$ defined by $df(p,v) = (f(p), d_pf(v))$ is smooth if $f:  M \rightarrow N$ is smooth.

 
Proof:  See Hitchin 4.1 (in Chapter 1 of\bh
http://www2.maths.ox.ac.uk/~hitchin/hitchinnotes/hitchinnotes.html

\eject

Defn: A {\it vector field} or {\it section of the tangent bundle} $TM$ is a smooth function
\bh
$s{:~}M\rightarrow TM$
so that $\pi\circ s={id}$ [i.e., $s(p) = (p, v_p)]$.  



Defn:  $s$ is {\it never zero} if $s(p) \not = (p, \0)$ for all $p \in M$.

Prop:  Let $G$ be a Lie group. Then $G$ admits a never-zero
vector field.

Note:  $S^n$ admits a never-zero
vector field iff $n$ odd.

Let $p_2(s(p)) = p_2(p, v_p) = v_p$

Defn:  The vector fields $s_1, ..., s_k$ are {\it linearly independent} iff for all $p 
\in M$, $p_2(s_1(p)), ..., p_2(s_k(p))$ are linearly independent.


Prop:  $M$ is parallelizable (or equivalently the \emph{``tangent 
bundle }$\pi$%
\emph{:~}$TM\rightarrow M$\emph{ is trivial''}) iff $TM$ admits $m$ 
linearly
independent vector fields.

Defn:  A {\it flow} on $M$ is a smooth
action of the Lie group ${\R}^{1}$ on $M$, $\sigma$:~${\R}%
^{1}\times M\rightarrow M$.

A flow is also called a {\it dynamical system}.
\vskip 10pt
\line{A {\it flow line} is the smooth path $\alpha_p: \R \rightarrow M$, 
$\alpha_p(t) = \sigma(t, p)$.}

\end



$G(p) = \{[g] ~|~ g^{smooth}:  U \rightarrow \R$, for some $U^{open}$ such that $p 
\in U
\subset M \}$

$C^{\infty}(M) =  \{g ~|~ g^{smooth}:  M \rightarrow \R \}$

$Z_a = \{g ~|~ g^{smooth}:  M \rightarrow \R, Dg_a = 0 \}$

The cotangent space $T^*_a = C^{\infty}(M)/ Z_a$. 

$T^*_a$ is an $m-$dimensional vector space:




$(df)_a