\magnification 1400
\parskip 10pt
\parindent 0pt
\hoffset -0.3truein
\hsize 7truein
\voffset -0.3truein
\vsize 10truein

\def\R{{\bf R}}
\def\C{{\bf C}}
\def\N{{\bf N}}
\def\r{{\bf r}}
\def\c{{\bf c}}
\def\p{{\bf p}}
\def\x{{\bf x}}
\def\y{{\bf y}}
\def\e{{\bf e}}
\def\i{{\bf i}}
\def\j{{\bf j}}
\def\k{{\bf k}}
\def\0{{\bf 0}}
\def\a{{\bf a}}
\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\l{\vskip 5pt
\hrule
\vskip -5pt}



Suppose $f:  N \rightarrow M$ is smooth.  Then $d_pf:  T_p(N) \rightarrow T_p(M)$.

If $df_{p}$ is $1-1$, for all $p\in N$, then 
$f$ is called an \emph{\it immersion}.

\centerline{I.e., $f$ is an immersion iff $f$ has rank $n$}

If $df_{p}$ is onto for all $p\in N$, then 
$f$ is called a \emph{\it submersion}.

\centerline{I.e., $f$ is an submersion iff $f$ has rank $m$}

Defn.  Suppose $f:  M \rightarrow N$ is smooth.  

$p \in M$ is a {\it critical point} and  $f(p)$ is a {\it critical value} if 
rank $df_p < n$.  

If $p \in M$ is not a critical point, then it is a {\it regular point}.  

If $q \in N$ is not a {\it critical value}, then it is a {\it regular value}. 

Note:  $q \in N$ is a regular value iff $f^{-1}(q) = \emptyset$ or $\forall p \in f^{-1}(q)$, $df_p = n$.

Defn:  If $K$ is a submanifold of $M$, then $K$ has the subspace topology. 
\hfil 
\break
Also, for all $p \in K$, there exists a chart $(\phi, U)$ 
for $M$ such that
$\phi(U) = \Pi_1^m(-\ep, \ep)$ and
 $\phi(U\cap K) = \Pi_1^k(-\ep, \ep) \times \Pi_{k+1}^m\{0\}$ 
, then \hfil \break
Also, $(\phi_{U \cap K }, U \cap K )$, $\phi|_{U 
\cap K }: U \cap K 
\rightarrow  \Pi_1^k(-\ep, \ep)$ is a chart for $K$.\bh  The collection of such charts form a 
pre-atlas 
for $K$.

\vfill

Thm 2.3.13:  Let $q$ be a regular value of $f$:~$M\rightarrow N$. Then
either $f^{-1}(q)=\emptyset $ or $f^{-1}(q)$ is an $(m-n)$-submanifold of $%
M$.

Defn:   Suppose $f$\emph{:}~$M\rightarrow N$ is a $1-1$ immersion,
and suppose $f$\emph{:}~$M\rightarrow f(M)$ is a homeomorphism, where $%
f(M)\subset N$ has the relative topology. Then $f$ is an \emph{\it embedding},
and $f(M)$ is an embedded submanifold. 

Thm.  $f:  M \rightarrow N$ embedding implies $f(M)$ is a submanifold of $N$.





 






\end
\l
Thm.  $f:  M \rightarrow N$ embedding implies $f(M)$ is a submanifold of $N$.

Recall $K$ is a submanifold of $N$ if $\forall q \in K \subset N$, $\exists g^{smooth}: V^{open} \subset N \rightarrow 
\R^{n-m}$, $q \in V$ such that $K \cap V = g^{-1}(0)$ and rank $d_p g = n-m$


Proof.  Since $f:  M \rightarrow N$ embedding,  $f:  M \rightarrow N$ is a 1-1 immersion and 

$f:M \rightarrow f(M)$ is a homeomorphism where $f(M)$ is a subspace of $N$ 

Take $q \in f(M)$.  

Since $f$ is 1:1, $\exists ! p \in M$ such that   $f(p) = q$.  

$f:  M \rightarrow N$ an immersion implies $f$ has rank $m \leq n$.


Thus by the rank theorem,

\l

Randell's Submanifolds (2.3) = Boothby's Regular submanifold (III.5):


Prop:  If $U^{open} \subset M^m$, then $U$ is an $m$-dimensional submanifold of 
$M$.

 
Prop:  If $K$ is a submanifold of $M$, then $i:  K \rightarrow M$, $i(k) = k$, 
the inclusion map is smooth.

Ex:  Find a counterexample to the above if we replace the hypothesis $K$ is a 
submanifold of $M$ with $K \subset M$.

Prop:  If $f:  N \rightarrow M$ is smooth and if  $H$ is a submanifold of $N$, 
then  
$f:  H \rightarrow M$ is smooth 

Ex:  Find a counterexample to the above if we replace the hypothesis $H$ is a 
submanifold of $N$ with $H \subset N$.


Prop:  If $f:  N \rightarrow M$ is smooth and if  $K$ is a submanifold of $M$ 
and if $f(N) \subset K$, then  
$f:  N \rightarrow K$ is smooth. 

Ex:  Find a counterexample to the above if we replace the hypothesis $K$ is a 
submanifold of $M$ with $K \subset M$.


\eject

Boothy III.6 = Randell Chapter 1.3

\vskip 5pt
\hrule
\vskip -5pt

Defn:  $G$ is a {\it topological group} if
\vskip -5pt
\h 1.) $(G, *)$ is a group

\h 2.)  $G$ is a topological space.

\h 3.)  $*: G \times G \rightarrow G$, $*(g_1, g_2) = g_1*g_2$, and

\leftline{~~~~~~~~$In: G \rightarrow G$, $In(g) = g^{-1}$ are both continuous 
functions.}


Defn:  $G$ is a {\it Lie group} if

\h 1.)  $G$ is a  group 

\h 2.)  $G$ is a smooth manifold.

\h 3.)  $*$ and $In$ are smooth functions.


Ex:  $Gl(n, \R) = $ set of all invertible $n \times n$ matrices is a Lie group:

\h 1.) $(Gl(n, \R), matrix ~multiplication)$ is a group

\h 2.)  $(Gl(n, \R) $ is a smooth manifold.

\h 3.)  $* (Gl(n, \R) \times  (Gl(n, \R) \rightarrow (Gl(n, \R)$, 
\h ~~~~~~~~~~~~~~~
$*(A, B) = AB$  and 
\h
~~~~$In: (Gl(n, \R) \rightarrow (Gl(n, \R)$ 
\h ~~~~~$In(A) = A^{-1}$ 
are smooth functions.

\l

Ex:  $(\C -\{\0\}, \cdot )$, is a Lie group.

Thm:  If $G$ is a Lie group and $H$ is a submanifold, then $H$ is a Lie group.

Ex:  $(S^1, \cdot )$

Ex:  $G_1, G_2$  
lie groups implies  $G_1 \times G_2$ is a lie group. 

Ex:  $T^n = S^1 \times ... \times S^1$ is a Lie group.

\vskip 5pt
\hrule
\vskip -5pt
The following maps are diffeomorphisms:  

$In:  G \rightarrow G$, $In(g) = g^{-1}$.

For $a \in G$,

$L_a: G \rightarrow G$, $L_a(g) = ag$

$R_a: G \rightarrow G$, $R_a(g) = ga$

Ex:  $O(n) = \{M \in GL(n, \R) ~|~ M^tM = I \} $ is a Lie group.


Ex: $Sl(n, \R) = \{M \in GL(n, \R) ~|~ det(M) = 1 \} $ is a Lie group.


Defn:  $F$ is a {\it homomorphism} of Lie groups if $F$ is an algebraic 
homomorphism of Lie groups and $F$ is smooth.


Ex:  $F: GL(n, \R) \rightarrow \R - \{\0\}$, $F(M) = det(M)$ is a homomorphism.

\l
Randell's Submanifolds (2.3) = Boothby's Regular submanifold (III.5):

$K \subset N$ is a $k$-submanifold of $N$ if $\forall p \in K$, there exists,

\l


Suppose $f: N \rightarrow M$ is smooth and has constant rank $k$.  If $q 
\in M$, then $f^{-1}(q)$ is a submanifold of $N$ of dimension $n - k$.

Proof:  Let $p \in f^{-1}(q)$. By the rank theorem,

\l


Ex:  $F: (\R, +) \rightarrow (S^1, \cdot)$, $F(t) = e^{2\pi it}$ is a 
homomorphism.

Ex:  $F: (\R^n, +) \rightarrow (T^n, \cdot)$, $F(t_1, ..., t_n) = (e^{2\pi 
it_1}, ..., e^{2\pi it_n}$ is a 
homomorphism.



Thm:  If $F:  G_1 \rightarrow G_2$ is a homomorphism of Lie groups, then 
\h
1.)  rank(F) is constant.
\h
2.)  kernel of $F$ = $F^{-1}(e)$ is a closed submanifold
\h
3.)  $F^{-1}(e)$ is a Lie group.
\h
4.)  $dim(ker~F) = dim(G_1) - rank(F)$

Thm:  If $H$ is a submanifold  and an algebraic subgroup of $G$, then $H$ is 
closed in $G$.

\vskip 5pt
\hrule
\vskip -5pt



Defn:  $G$ = group, $X =$ set.
{\it $G$ acts on $X$} (on the left) if $\exists \s:  G \times X \rightarrow X$ 
such 
that

\h 1.)  $\s(e, x) = x ~~\forall x \in X$

\h 2.)  $\s(g_1, \s(g_2, x)) = \s(g_1g_2, x)$

Notation: $\s(g, x) = gx$.  \hfil \break
Thus 1) $ex = x$; 2) $g_1(g_2x) = (g_1g_2)(x)$.

\u
If $G$ is a Lie group and $X$ is a smooth manifold, then we require $\s$ to be  
smooth.


Defn:  The {\it orbit} of $x \in X$ = \hfil \break
$G(x) = \{ y \in X ~|~ \exists g$ such that $y = gx\}$

Note: \hfil \break
 1.)  $x \in G(x)$ \hfil %%\break
 2.)  If $G(x) \cap G(y) \not= \emptyset$, then $G(x) = G(y)$

Thus we can use an action of $G$ to partition $X$ into disjoint subsets.

Defn:  If $G$ acts on $X$, then  $X/G = X/\sim$ where  $x \sim y$ iff $y \in 
G(x)$ iff $\exists g$ such that $y = gx$.

If $X$ is a topological space,  then $X/G = X/\sim$ is a topological space with 
the quotient topology.

When is $X/G = X/\sim$ a manifold?



Ex:  $G= ({\bf Z}, +)$, $M = \R$, $\s(n, x) = n + x$.

$M/G =$

Ex:  $G= ({\bf Z} \times {\bf Z}, +)$, $M = \R^2$, \hfil \break $\s((n, 
m), (x,y)) = 
(n + x, m + y)$.

$M/G =$

Ex:  $G= ({\bf Z_2}, +)$, $M = S^n$, $\s(0, x) = x$, $\s(1, x) = -x$, .

$M/G =$

\end
\eject

Defn:  The action of $G$ on $X$ is {\it free } if $gx = x$ implies $g = e$.

Thm 1.3.9:  If $M$ is a smooth $n$-manifold, and $G$ is a finite Lie group 
acting freely on $M$, then $M/G$ is a smooth $n$-manifold.  Also, $p:  M 
\rightarrow M/G$ is smooth.


Defn:  $G$ is a {\it discrete group} if  
  \vskip -5pt
\h 0.)  $G$ is a group.
\vskip -5pt
\h 1.)  $G$ is countable
\vskip -5pt
\h 2.)  $G$ has the discrete topology

Note a discrete group is a Lie group.



Defn:  The action of $G$ on $M$ is {\it properly discontinuous} if $\forall x 
\in 
M$, $\exists U^{open}$ such 
that $x \in U$ and $U \cap gU = \emptyset ~~\forall g \in G$.


\vskip 5pt
\hrule
\vskip -5pt


Ex:  $({\bf Z}, +)$ acting on $\R^1$ where $\s(n, x) = n + x$.

Thm 1.3.2:  $M$ smooth $n$-manifold, $G$ discrete group acting properly 
discontinuously on $M$ implies $M/G$ is a smooth $n$-manifold.  Also, $p:  M 
\rightarrow M/G$ is smooth.




\end

3, 6, 7

\eject




$\matrix{
 & f &&  &  \cr
M & \longrightarrow  &  f(M) & \subset  & N\cr
\cup  &  &&&  \cup \cr
U  &  & && V \cr
\downarrow  \phi^{homeo}&&&  & \psi^{homeo} \downarrow \cr
 & \psi \circ f \circ \phi = \pi_(1...m) &&   &  \cr
\Pi_1^m (-\ep, \ep) & \longrightarrow   &&    & \Pi_1^n (-\ep, \ep) )\cr
}$


\end