\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{differential geometry}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{differential_geometry}{}\paragraph*{{Differential geometry}}\label{differential_geometry}

[[!include synthetic differential geometry - contents]]

\hypertarget{higher_geometry}{}\paragraph*{{Higher geometry}}\label{higher_geometry}

[[!include higher geometry - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{scope}{Scope}\dotfill \pageref*{scope} \linebreak
\noindent\hyperlink{generalized_smooth_spaces_from_pov}{Generalized smooth spaces from $n$POV}\dotfill \pageref*{generalized_smooth_spaces_from_pov} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{References}{References}\dotfill \pageref*{References} \linebreak
\noindent\hyperlink{traditional}{Traditional}\dotfill \pageref*{traditional} \linebreak
\noindent\hyperlink{higher}{Higher}\dotfill \pageref*{higher} \linebreak
\noindent\hyperlink{derived}{Derived}\dotfill \pageref*{derived} \linebreak


\hypertarget{scope}{}\subsection*{{Scope}}\label{scope}

\textbf{Differential geometry} is a mathematical discipline studying geometry of spaces using differential and integral calculus. Classical differential geometry studied submanifolds (curves, surfaces\ldots{}) in Euclidean spaces. The traditional objects of differential geometry are finite and infinite-dimensional [[differentiable manifold]]s modelled locally on [[topological vector space]]s. Techniques of differential calculus can be further stretched to [[generalized smooth space]]s. One often distinguished analysis on manifolds from differential geometry: analysis on manifolds focuses on functions from a manifold to the ground field and their properties, togehter with applications like PDEs on manifolds. Differential geometry on the other hand studies objects embedded into the manifold like submanifolds, their relations and additional structures on manifolds like bundles, connections etc. while the topological aspects are studied in a younger branch (from 1950s on) which is called [[differential topology]].

\hypertarget{generalized_smooth_spaces_from_pov}{}\subsection*{{Generalized smooth spaces from $n$POV}}\label{generalized_smooth_spaces_from_pov}

See also [[generalized smooth space]].

Finite-dimensional \emph{differential geometry is the [[geometry]] modeled on [[Cartesian space]]s and [[smooth functions]] between them.}

Formally, it is the [[higher geometry|geometry]] modeled on the [[pregeometry (for structured (infinity,1)-toposes)|pre-geometry]] $\mathcal{G} =$[[CartSp]].

This includes a sequence of concepts of [[generalized smooth space]]s:

\begin{itemize}%
\item A [[smooth manifold]] (see there for details) is a locally $CartSp$-representable object in the [[sheaf topos]] $Sh(CartSp)$.


\item A [[diffeological space]] (see there) is a [[concrete sheaf]] in the [[cohesive topos]] $Sh(CartSp)$.


\item a [[Lie groupoid]] is a locally representable object in the [[(2,1)-sheaf]] [[(2,1)-topos]] $Sh_{(2,1)}(CartSp)$;


\item an [[∞-Lie groupoid]] is an object in the [[(∞,1)-sheaf (∞,1)-topos]] $Sh_{(\infty,1)}(CartSp)$.



\end{itemize}
Similarly, standard models of [[synthetic differential geometry]] in [[higher geometry]] are modeled on the [[pregeometry (for structured (infinity,1)-toposes)|pre-geometry]] $\mathcal{G} =$[[ThCartSp]]. To wit, the [[cohesive topos]] $Sh(ThCartSp)$ is the [[smooth topos]] called the [[Cahiers topos]]:

\begin{itemize}%
\item an [[infinitesimal space]] is a certain object in $ThCartSp$;


\item an [[∞-Lie algebroid]] is a certain object in the [[(∞,1)-category of (∞,1)-sheaves]] $Sh_{(\infty,1)}(ThCartSp)$.



\end{itemize}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[synthetic differential geometry]]


\item [[higher differential geometry]], [[derived differential geometry]]

\begin{itemize}%
\item [[higher differential geometry applied to plain differential geometry]]

\end{itemize}

\item [[prequantum geometry]], [[higher prequantum geometry]]


\item [[D-geometry]]


\item [[arithmetic differential geometry]]


\item [[Lie theory]]


\item [[fiber bundles in physics]]


\item some modern subfields of differential geometry include:

\begin{itemize}%
\item [[symplectic geometry]],


\item [[contact geometry]],


\item [[complex geometry]],


\item [[Riemannian geometry]],


\item [[Finsler geometry]],


\item [[symmetric spaces]],


\item [[Fréchet manifolds]]



\end{itemize}


\end{itemize}
$\,$

\begin{tabular}{l|l}
local model&global geometry\\
\hline 
[[Klein geometry]]&[[Cartan geometry]]\\
[[Klein 2-geometry]]&[[Cartan 2-geometry]]\\
[[higher Klein geometry]]&[[higher Cartan geometry]]\\
\end{tabular}

$\,$

[[!include geometries of physics -- table]]

\hypertarget{References}{}\subsection*{{References}}\label{References}

\hypertarget{traditional}{}\subsubsection*{{Traditional}}\label{traditional}

The study of differential geometry goes back to the study of [[surfaces]] [[embedding of smooth manifolds|embedded]] into [[Euclidean space]] $\mathbb{R}^3$ in

\begin{itemize}%
\item [[Carl Friedrich Gauss]], \emph{General Investigations of Curved Surfaces}, 1827 (\href{http://www.gutenberg.org/ebooks/36856}{Gutenberg})

\end{itemize}
Textbooks include

\begin{itemize}%
\item [[Shoshichi Kobayashi]], [[Katsumi Nomizu]], \emph{Foundations of differential geometry} , Volume 1 (1963), Volume 2 (1969), Interscience Publishers, reprinted 1996 by Wiley Classics Library


\item [[Michael Spivak]], \emph{A comprehensive introduction to differential geometry} (5 Volumes)


\item [[Michael Spivak]], \emph{[[Calculus on Manifolds]]} (1971)


\item [[M M Postnikov]], \emph{Lectures on geometry} (6 vols.: 1 ``Analytic geometry'', 2 ``Linear algebra'', 3 ``Diff. manifolds''; 4 ``Diff. geometry'' (covers extensively fibre bundles and connections); 5 ``Lie groups''; 6 ``Riemannian geometry'')



\end{itemize}
With emphasis in [[G-structures]]:

\begin{itemize}%
\item [[Shlomo Sternberg]], \emph{Lectures on differential geometry}, Prentice-Hall (1964)

\end{itemize}
With emphasis on [[Cartan geometry]]:

\begin{itemize}%
\item R. Sharpe, \emph{Differential geometry -- Cartan's generalization of Klein's Erlagen program}, Springer (1997)

\end{itemize}
Lecture notes include

\begin{itemize}%
\item [[Brian Conrad]], Handouts on Differential Geometry (\href{http://math.stanford.edu/~conrad/diffgeomPage/handouts.html}{web})


\item [[Liviu Nicolaescu]], \emph{Lectures on the Geometry of Manifolds}, 2018 (\href{https://www3.nd.edu/~lnicolae/Lectures.pdf}{pdf})



\end{itemize}
An introduction with an eye towards applications in [[physics]], specifically to [[gravity]] and [[gauge theory]] is in

\begin{itemize}%
\item [[Theodore Frankel]], \emph{[[The Geometry of Physics - An Introduction]]}


\item [[Chris Isham]], \emph{[[Modern Differential Geometry for Physicists]]}



\end{itemize}
A discussion in the context of [[Frölicher spaces]] and [[diffeological spaces]] is in

\begin{itemize}%
\item [[Andreas Kriegl]], [[Peter Michor]], \emph{[[The convenient setting of global analysis]]}, Math. Surveys and Monographs \textbf{53}, Amer. Math. Soc. 1997. 618 pages

\end{itemize}
Discussion with emphasis on [[natural bundles]] is in

\begin{itemize}%
\item [[Ivan Kolá?]], [[Peter Michor]], [[Jan Slovák]], \emph{[[Natural operators in differential geometry]]} (\href{http://www.emis.de/monographs/KSM/kmsbookh.pdf}{pdf})

\end{itemize}
See also

\begin{itemize}%
\item Sigmundur Gudmundsson, \emph{An Introduction to Riemannian Geometry} (\href{http://www.maths.lth.se/matematiklu/personal/sigma/Riemann.pdf}{pdf})


\item Wikipedia, \emph{\href{http://en.wikipedia.org/wiki/Differential_geometry}{differential geometry}}



\end{itemize}
\hypertarget{higher}{}\subsubsection*{{Higher}}\label{higher}

See at \emph{[[higher differential geometry]]}.

\hypertarget{derived}{}\subsubsection*{{Derived}}\label{derived}

For [[derived differential geometry]] see

\begin{itemize}%
\item [[Dominic Joyce]], \emph{D-manifolds and d-orbifolds: a theory of derived differential geometry} (\href{http://people.maths.ox.ac.uk/joyce/dmanifolds.html}{web})


\item [[Urs Schreiber]], \emph{[[schreiber:Seminar on derived differential geometry]]}



\end{itemize}


\end{document}