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%%tth:\special{html:<title> A very brief introduction to differentiable manifolds</title>}

%Slide 1
\title{{\LARGE\bf A very brief introduction to differentiable manifolds}\newline \newline \newline}
\author{Tom Carter
\newline
\newline
\newline
\tthdump{\href
{http://cogs.csustan.edu/\~tom/diff-manifolds}{http://cogs.csustan.edu/\~\ tom/diff-manifolds}}
%%tth:\href{http://cogs.csustan.edu/~tom/diff-manifolds}{http://cogs.csustan.edu/\~tom/diff-manifolds}
\vfill
Santa Fe Institute\newline
Complex Systems Summer School
\newline
}

\date{June, 2001}


\maketitle

%Slide 2
\sectionhead{Our general topics:}
%%tth:\begin{itemize}
%%tth:\item
\tthdump{\hyperlink{Why differentiable manifolds}
               {\ $\circledcirc$ Why differentiable manifolds\newline}}
%%tth:\makehyperlink{Why differentiable manifolds}
%%tth:\item
%%% \tthdump{\hyperlink{Topological spaces}
%%% 	        {$\circledcirc$ Topological spaces\newline}} 
%%% %%tth:\makehyperlink{Topological spaces}
\tthdump{\hyperlink{Topological spaces}
	        {$\circledcirc$ Topological spaces}}
\tthdump{\hyperlink{Topological spaces.ex}
	        { (ex)\newline}} 
%%tth:\makehyperlink{Topological spaces}
%%tth:\makehyperlinkex{Topological spaces.ex}
%%tth:\item
\tthdump{\hyperlink{Examples of topological spaces}
	        {$\circledcirc$ Examples of topological spaces}} 
\tthdump{\hyperlink{Examples of topological spaces.ex}
	        { (ex)\newline}}
%%tth:\makehyperlink{Examples of topological spaces}
%%tth:\makehyperlinkex{Examples of topological spaces.ex}
%%tth:\item
\tthdump{\hyperlink{Coordinate systesm and manifolds}
	        {$\circledcirc$ Coordinate systesm and manifolds}} 
\tthdump{\hyperlink{Coordinate systesm and manifolds.ex}
	        { (ex)\newline}} 
%%tth:\makehyperlink{Coordinate systesm and manifolds}
%%tth:\makehyperlinkex{Coordinate systesm and manifolds.ex}
%%tth:\item
\tthdump{\hyperlink{Manifolds}
	        {$\circledcirc$ Manifolds}} 
\tthdump{\hyperlink{Manifolds.ex}
	        { (ex)\newline}} 
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%%tth:\makehyperlinkex{Manifolds.ex}
% %%tth:\item
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% 	        { (ex)\newline}} 
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% %%tth:\item
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% 	        { (ex)\newline}} 
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% %%tth:\item
% \tthdump{\hyperlink{Linear transformations}
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% \tthdump{\hyperlink{Linear transformations.ex}
% 	        { (ex)\newline}} 
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% %%tth:\item
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% 	        { (ex)\newline}} 
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% %%tth:\makehyperlinkex{Morphisms -- mono, epi, and iso.ex}
% %%tth:\item
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% \tthdump{\hyperlink{Linear operators.ex}
% 	        { (ex)\newline}} 
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% %%tth:\item
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% 	         {$\circledcirc$ Normed linear spaces}} 
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% 	        { (ex)\newline}} 
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% %%tth:\item
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% 	        { (ex)\newline}} 
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% %%tth:\item
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% 	        { (ex)\newline}} 
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% %%tth:\item
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% 	        { (ex)\newline}} 
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%%  %%tth:\item
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%%  	        { (ex)\newline}} 
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%%tth:\item
\tthdump{\hyperlink{References}
	        {$\circledcirc$ References\newline}} 
%%tth:\makehyperlink{References}
%%tth:\end{itemize}

(ex):  exercises.

\pagedone

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% \pagedone
% 
% %Slide 3
% 
% \quotesection{Science, wisdom, and counting}
% %%tth:\begin{quote}
% ``Science is organized knowledge. Wisdom is organized life.''
% 
% - Immanuel Kant 
% 
% ``My own suspicion is that the universe is not only stranger than we suppose, but stranger than we can s% uppose.''
% 
% - John Haldane
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% - Edward Gibbon 
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% %%tth:\end{quote}
% 
% \pagedone

\sectionhead{Why differentiable manifolds}

\begin{itemize}
       \item Differentiable manifolds can generally be thought of as a generalization of
            $\myreal^n$.  They are mathematical objects equipped with smooth (local)
            coordinate systems.  Much of physics can be thought of as having a natural
            home in differentiable manifolds.  A particularly valuable aspect of
            differentiable manifolds is that unlike traditional flat (Euclidean)
            $\myreal^n$, they can have (intrinsic) curvature.

\end{itemize}
 
 \pagedone
  
\sectionhead{Topological spaces}

\begin{itemize}
	\item We need a way to talk about ``nearness'' of points in a
	     space, and continuity of functions.  We can't (yet) talk about the ``distance''
	     between pairs of points or limits of sequences -- we will use a more
	     abstract approach.  We start with:
	     
	     {\bf Def.:}  A {\bf topological space} $(\mathbf{X}, \mathbf{T})$ is a set
	     $\mathbf{X}$ together with a topology $\mathbf{T}$ on $\mathbf{X}$.  A topology
	     on a set $\mathbf{X}$ is a collection of subsets of $\mathbf{X}$ (that is,
	     $T \subset \mathcal{P}(\mathbf{X})$) satisfying:
\begin{enumerate}
		\item If $G_1, G_2 \in \mathbf{T}$, then $G_1 \cap G_2 \in \mathbf{T}$.
		\item If $\{G_\alpha\ |\ \alpha \in J\}$ is any collection of sets in $\mathbf{T}$,
		     then
		     $$\bigcup_{\alpha \in J}G_\alpha \in \mathbf{T}.$$
		\item  $\emptyset \in \mathbf{T}$, and $\mathbf{X} \in \mathbf{T}$.
	\end{enumerate}	

     \item  The sets $G \in \mathbf{T}$ are called {\em open sets} in $\mathbf{X}$.  A
          subset $F \subset \mathbf{X}$ whose complement is open is called a
          {\em closed set} in $\mathbf{X}$.
          
     \item  If $A$ is any subset of a topological space $X$, then the {\em interior} of
          $A$, denoted by $A^\circ$, is the union of all open sets contained in $A$.
          The {\em closure} of $A$, denoted by $\overline{A}$, is the intersection
          of all closed sets containing $A$.
          
     \item  If $x \in X$, then a {\em neighborhood} of $x$ is any subset $A \subset X$
          with $x \in A^\circ$.
          
     \item  If $(X, T)$ is a topological space, and $A$ is a subset of $X$, then
          the {\em induced} or {\em subspace} topology $T_A$ on $A$ is given by
          $$T_A = \{G \cap A\ |\ G \in T\}.$$
          It is easy to check that $T_A$ actually is a topology on $A$.  With this
          topology, $A$ is called a {\em subspace} of $X$.
          
     \item  Suppose $X$ and $Y$ are topological spaces, and $f : X \to Y$.  Recall
          that if $V \subset Y$, we use the notation
          $$f^{-1}(V) = \{x \in X\ |\ f(x) \in V\}.$$
          We then have the definition:
          
          {\bf Def.:}  A function $f : X \to Y$ is called {\em continuous} if
          $f^{-1}(G)$ is open in $X$ for every open set $G$ in $Y$.
          
     \item  We can also define continuity at a point.  Suppose $f : X \to Y$,
          $x \in X$, and $y = f(x)$.  We say that $f$ is {\em continuous at
          x} if for every neighborhood $V$ of $y$, there is a neighborhood
          $U$ of $x$ with $f(U) \subset V$.  We then say that a function
          $f$ is {\em continuous} if it is continuous at every $x \in X$.
          
     \item  A {\em homeomorphism} from a topological space $X$ to a topological space $Y$
          is a 1-1, onto, continuous function $f : X \to Y$ whose inverse is also continous.
          
     \item  A topological space is called {\em separable} if there is a countable collection
          of open sets such that every open set in $T$ can be written as a union of
          members of the countable collection.
          
     \item  A topological space $X$ is called {\em Hausdorff} if for every $x, y \in X$
          with $x \ne y$, there are neighborhoods $U$ and $V$ of $x$ and $y$ (respectively)
          with $U \cap V = \emptyset$. 
          
     \item  This is just the barest beginnings of Topology, but it should be enough
          to get us off the ground \ldots
	
	\end{itemize}

 \pagedone
 
 \exercises{Topological spaces}
 
 \begin{enumerate}
      \item Show that the intersection of a finite number of open sets is open.  Give
           an example to show that the intersection of an infinite number of open
           sets may not be open.
      \item How many distinct topologies are there on a set containing three elements?
      \item Show that the interior of a set is open.  Show that the closure of a set
           is closed.  Show that $A^\circ \subset A \subset \overline{A}$.  Show
           that it is possible for $A^\circ$ to be empty even when $A$ is not empty.
      \item Show that if $f : X \to Y$ is continuous, and $F \subset Y$ is closed,
           then $f^{-1}(F)$ is closed in X.
      \item Show that a set can be both open and closed.  Show that a set can be
           neither open nor closed.
      \item  Show that if $f : X \to Y$ and $g : Y \to Z$ are both continuous,
           then $g \circ f : X \to Z$ is continuous.
      \item  Show that the two definitions of continuity are equivalent.
      \item  A subset $D \subset X$ is called {\em dense} in $X$ if $\overline{D} = X$.
           Show that it is possible to have a dense subset $D$ with $D^\circ = \emptyset$.
      \item  Show that if $D$ is dense in $X$, then for every open set $G \subset X$,
           we have $G \cap D \ne \emptyset$.  In particular, every neighborhood of every
           point in $X$ contains points in $D$.
      \item  Show that in a Hausdorff space, every set consisting of a single point $x$
           (i.e., $\{x\}$) is a closed set.
 \end{enumerate}
 
 \pagedone
 
 \sectionhead{Examples of topological spaces}
 
 \begin{itemize}
        \item For any set $X$, there are two trivial topologies:
             $$T_c = \{\emptyset, X\}$$
             and
             $$T_d = \mathcal{P}(X).$$
             $T_d$ is the topology in which each point (considered as a subset)
             is open (and hence, every subset is open).  It is called the {\em discrete}
             topology.  $T_c$ is sometimes called the {\em concrete} topology.
        
        \item  On $\myreal$, there is the usual topology.  We start with open
             intervals $(a, b) = \{x\ |\ a < x < b\}$.  An open set is then any
             set which is a union of open intervals.
        
        \item  On $\myreal^n$, there is the usual topology.  One way to get this is to
             begin with the open balls with center $a$ and radius $r$, where $a \in \myreal^n$
             can be any point in $\myreal^n$, and $r$ is any positive real number:
             $$B_n(a, r) = \{x \in \myreal^n\ |\ |x - a| < r\}.$$
             An open set is then any set which is a union of open balls.
             
                    
\end{itemize}
 \pagedone
 
 \exercises{Examples of topological spaces}
 
 \begin{enumerate}
 
      \item  Check that each of the examples actually is a topological space.
      
      \item  For $k < n$, we can consider $\myreal^k$ to be a subset of $\myreal^n$.
           Show that the inherited subspace topology is the same as the usual
           topology.
           
      \item  Show that $\myreal^n$ with the usual topology is separable and Hausdorff.
           
 \end{enumerate}
 
 \pagedone
 
 \sectionhead{Coordinate systems and manifolds}
 
 \begin{itemize}
      \item  Suppose $M$ is a topological space, $U$ is an open subset of $M$,
           and $\mu : U \to \myreal^n$.  Suppose further that $\mu(U)$ is an open
           subset of $\myreal^n$, and that $\mu$ is a homeomorphism between $U$ and $\mu(U)$.
           We call $\mu$ a {\em local coordinate system} of dimension $n$ on $U$.
           
           For each point $m \in U$, we then have that $\mu(m) = (\mu_1(m), \ldots ,
           \mu_n(m))$, the coordinates of $m$ with respect to $\mu$.
           
      \item  Now suppose that we have another open subset $V$ of $M$, and $\nu$ is
           a local coordinate system on $V$.  We say that $\mu$ and $\nu$ are $C^infty$
           {\em compatible} if the composite functions $\mu \circ \nu^{-1}$
           and $\nu \circ \mu^{-1}$
           are $C^\infty$ functions on $\mu(U) \cap \nu(V)$.  Remember that a function on
           $\myreal^n$ is $C^\infty$ if it is continuous, and all its partial
           derivatives are also continuous.
      
      \item  A {\em topological manifold} of dimension $n$ is a separable Hausdorff
           space $M$ such that every point in $M$ is in the domain of a local coordinate
           system of dimension $n$.  These spaces are sometimes called {\em locally
           Euclidean} spaces.
           
      \item  A $C^\infty$ {\em differentiable structure} on a topological manifold $M$ is a
           collection $\mathcal{F}$ of local coordinate systems on $M$ such that:
           
           \begin{enumerate}
                \item  The union of the domains of the local coordinate systems is
                     all of $M$.
                \item  If $\mu_1$ and $\mu_2$ are in $\mathcal{F}$, then
                     $\mu_1$ and $\mu_2$ are $C^\infty$ compatible.
                \item  $\mathcal{F}$ is maximal with respect to 2.  That is,
                     if $\nu$ is $C^\infty$ compatible with all $\mu \in \mathcal{F}$,
                     then $\nu \in \mathcal{F}$.
           \end{enumerate}
           
      \item  A $C^\infty$ {\em differentiable manifold} of dimension $n$ is a
           topological manifold $M$ of dimension $n$, together with a $C^\infty$
           differentiable structure $\mathcal{F}$ on $M$.
           
           Notes:
           \begin{enumerate}
                \item  It is possible for a topological manifold to have more than
                     one distinct differentiable structures.
                \item  In this discussion, we have limited ourselves to $C^\infty$
                     differentiable structures.  With somewhat more work, we could
                     define $C^k$ structures for $k < \infty$.
                \item  We have limited the domains of our local coordinate systems
                     to be open subsets of M.  This means that the usual spherical
                     and cylindrical coordinate systems on $\myreal^3$ do not count
                     as local coordinate systems by our definition.
                \item  With somewhat more work, we could define differentiable
                     manifolds with boundaries.
                \item  We have limited ourselves to manifolds of finite dimension.
                     With somewhat more work, we could define infinite dimensional
                     differentiable manifolds.
           \end{enumerate}

\pagedone

 \end{itemize}
 
\pagedone

\exercises{Coordinate systems and manifolds}

\begin{enumerate}

     \item  
          
\end{enumerate}

\pagedone

\sectionhead{Manifolds}


\begin{itemize}
       \item 
       
\end{itemize}

\pagedone

\exercises{Manifolds}

\begin{enumerate}

     \item  
     
\end{enumerate}

\pagedone



 \footnotesize
 \bibliographystyle{plain} 
 
\tthdump{\hypertarget{References}{}\hyperlink{Our general topics:}{\hfil To top $\leftarrow$}}
%%tth:{\special{html: <A NAME="References"></A><a href="\#Top of file">       Top</a>}}
 
 \begin{thebibliography}{12}
%%tth:{\special{html: <font size="+0">}}

\bibitem{auslander}
Auslander, Louis, and MacKenzie, Robert E.,
{\em Introduction to Differentiable Manifolds},
Dover Publications, New York, 1977.

\bibitem{bishop}
Bishop, Richard L., and Goldberg, Samuel I.,
{\em Tensor Analysis on Manifolds},
Dover Publications, New York, 1980.


\bibitem{warner}
Warner, Frank W.,
{\em Foundations of Differentiable Manifolds and Lie Groups},
Scott, Foresman and Company, Glenview, Illinois, 1971.


\end{thebibliography}

\tthdump{\hyperlink{Our general topics:}{\hfil To top $\leftarrow$}}
%%tth:{\special{html: <a href="\#Top of file">       Back to top of file</a>}}

% 
% \sectionhead{On-line references}
% 
% 
% Some of the references listed above are available on line.  They are listed again here for easy access:
% 
% %\bibitem{abrams2}
% Abrams D S and Lloyd S, 
% Non-Linear Quantum Mechanics implies Polynomial Time 
% solution for NP-complete and $\#$P problems,
% %in {\it LANL e-print} quant-ph/9801041, http://xxx.lanl.gov (1998)
% \hyperref{http://xxx.lanl.gov/abs/quant-ph/9801041}{}{}
% %\hyperURL{http}{xxx.lanl.gov/abs/quant-ph}{9801041}
% {http://xxx.lanl.gov/abs/quant-ph/9801041}
% 
% 
% %\bibitem{aharonov5}
% Aharonov D, Beckman D, Chuang I and  Nielsen M,
% What Makes Quantum Computers Powerful? 
% \hyperref{http://wwwcas.phys.unm.edu/\~mnielsen/science.html}{}{}
% %\hyperURL{http}{wwwcas.phys.unm.edu/~mnielsen}{science.html}
% {http://wwwcas.phys.unm.edu/\~mnielsen/science.html}
% % 
% % 
% 
% %\bibitem{decoherence2}
%  Chuang I L, Laflamme R and Paz J P, 
% Effects of Loss and Decoherence on a Simple Quantum Computer,
% %in {\it LANL e-print} quant-ph/9602018,  http://xxx.lanl.gov (1996)
% \hyperref{http://xxx.lanl.gov/abs/quant-ph/9602018}{}{}
% %\hyperURL{http}{xxx.lanl.gov/abs/quant-ph}{9602018}
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% 

\end{document}
