\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*{calibration}

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

\hypertarget{riemannian_geometry}{}\paragraph*{{Riemannian geometry}}\label{riemannian_geometry}

[[!include Riemannian geometry - contents]]

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

[[!include synthetic differential geometry - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{minimal_volume_submanifolds}{Minimal volume submanifolds}\dotfill \pageref*{minimal_volume_submanifolds} \linebreak
\noindent\hyperlink{CalibrationsFromSpinors}{Calibrations from spinors}\dotfill \pageref*{CalibrationsFromSpinors} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{application_in_string_theory}{Application in string theory}\dotfill \pageref*{application_in_string_theory} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A degree-$p$ \emph{calibration} of an oriented [[Riemannian manifold]] $(X,g)$ is a [[differential p-form]] $\omega \in \Omega^p(X)$ with the property that

\begin{enumerate}%
\item it is [[closed differential form|closed]] $d \omega = 0$;


\item evaluated on any oriented $p$-dimensional subspace of any [[tangent space]] of $X$, it is less than or equal to the induced degree-$p$ [[volume form]], with [[equality]] for at least one choice of subspace.



\end{enumerate}
A Riemannian manifold equipped with such a calibration is also called a \emph{calibrated geometry} (\hyperlink{HarveyLawson82}{Harvey-Lawson 82}) or similar.

A \emph{calibrated submanifold} of a manifold with calibration is an oriented [[submanifold]] such that restricted to each of its tangent spaces $\omega$ \emph{equals} the induced volume form of the submanifold there.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{minimal_volume_submanifolds}{}\subsubsection*{{Minimal volume submanifolds}}\label{minimal_volume_submanifolds}

Any calibrated submanifold $\Sigma \hookrightarrow X$ minimizes [[volume]] in its [[homology]] class.

For Let $\tilde \Sigma \hookrightarrow X$ be a homologous [[submanifold]]. Then [[Stokes theorem]] together with the condition that $d \phi = 0$ implies that the [[integration of differential forms]] of $\phi$ over $\Sigma$ equals that over $\tilde \Sigma$. The defining conditions on calibrations and on calibrated submanifolds then imply the [[inequality]]

\begin{displaymath}
vol(\Sigma)
  \stackrel{cal\,subm}{=}
  \int_\Sigma \phi
  \stackrel{Stokes}{=} 
  \int_{\tilde \Sigma} \phi
  \stackrel{calib}{\leq}
  \int_{\tilde \Sigma} d vol
  =
  vol(\tilde \Sigma)
  \,.
\end{displaymath}
\hypertarget{CalibrationsFromSpinors}{}\subsubsection*{{Calibrations from spinors}}\label{CalibrationsFromSpinors}

\begin{quote}%
under construction


\end{quote}
For suitable $n$ and $p$, and given a real [[spin representation]] of $Spin(n)$, then the [[Cartesian space]] $\mathbb{R}^n$ with its canonical Riemannian structure becomes $p$-calibrated with the calibration form being

\begin{displaymath}
\omega_{\epsilon}
  \coloneqq 
  (\overline{\epsilon} \Gamma_{a_1 a_2 \cdots a_p} \epsilon) \, e^{a_1} \wedge \cdots \wedge e^{a_p}
\end{displaymath}
where

\begin{enumerate}%
\item $\{e^a\}$ denotes the canonical [[linear basis]] of [[differential 1-forms]];


\item $\epsilon$ is a non-vanishing [[spinor]];


\item $\overline{\epsilon} \Gamma_{a_1 a_2 \cdots a_p} \epsilon$ is the \href{spin+representation#SpinorBilinearForms}{canonical bilinear pairing} which in components is given by evaluating $\epsilon$ in the [[quadratic form]] given by multiplying the skew-symmetrized product of $p$ of the representation matrices $\Gamma^a$ of the [[Clifford algebra]] with the [[charge conjugation matrix]] $C$.



\end{enumerate}
(e.g. \hyperlink{DadokHarvey93}{Dadok-Harvey 93}).

For instance for $n = 7$ and $p = 3$ then this gives the [[associative 3-form]] calibration.

More generally for $X$ an $n$-dimensional [[Riemannian manifold]] with a [[covariantly constant spinor]] $\epsilon$, then under suitable conditions applying this construction in each [[tangent space]] gives a calibration.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item The globalization of the [[associative 3-form]] of a [[G2-manifold]] is a calibration. A calibrated submanifold in this case is also called an [[associative submanifold]].


\item The [[Cayley 4-form]] on [[Spin(7)-manifolds]].



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

The original articles are

\begin{itemize}%
\item [[Reese Harvey]], [[H. Blaine Lawson]], \emph{Calibrated geometries}, Acta Mathematica July 1982, Volume 148, Issue 1, pp 47-157


\item [[Reese Harvey]], \emph{Calibrated geometries}, Proceeding of the ICM 1983 (\href{http://www.mathunion.org/ICM/ICM1983.1/Main/icm1983.1.0797.0808.ocr.pdf}{pdf})



\end{itemize}
The relation to [[Killing spinors]] goes back to

\begin{itemize}%
\item [[Reese Harvey]], \emph{Spinors and Calibrations}, Academic Press, 1990 (\href{http://www.elsevier.com/books/spinors-and-calibrations/harvey/978-0-12-329650-4}{publisher})


\item [[Jiri Dadok]], [[Reese Harvey]], \emph{Calibrations and spinors}, Acta Mathematica 1993, Volume 170, Issue 1, pp 83-120



\end{itemize}
See also

\begin{itemize}%
\item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Calibrated_geometry}{Calibrated geometry}}


\item Jason Dean Lotay, \emph{Calibrated submanifolds and the Exceptional geometries}, 2005 (\href{https://people.maths.ox.ac.uk/joyce/theses/LotayDPhil.pdf}{pdf})



\end{itemize}
\hypertarget{application_in_string_theory}{}\subsubsection*{{Application in string theory}}\label{application_in_string_theory}

Discussion in [[string theory]]/[[M-theory]] includes the following.

\begin{itemize}%
\item [[Gary Gibbons]], [[George Papadopoulos]], \emph{Calibrations and Intersecting Branes} (\href{http://arxiv.org/abs/hep-th/9803163}{arXiv:hep-th/9803163})


\item [[Jerome Gauntlett]], [[Neil Lambert]], [[Peter West]], \emph{Branes and Calibrated Geometries}, Commun.Math.Phys. 202 (1999) 571-592 (\href{http://arxiv.org/abs/hep-th/9803216}{arXiv:hep-th/9803216})


\item [[George Papadopoulos]], [[Jan Gutowski]], \emph{AdS Calibrations}, Phys.Lett.B462:81-88,1999 (\href{http://arxiv.org/abs/hep-th/9902034}{arXiv:hep-th/9902034})


\item [[Jan Gutowski]], [[George Papadopoulos]], [[Paul Townsend]], \emph{Supersymmetry and generalized calibrations}, Phys.Rev.D60:106006, 1999 (\href{http://arxiv.org/abs/hep-th/9905156}{arXiv:hep-th/9905156})


\item [[Jan Gutowski]], S. Ivanov, [[George Papadopoulos]], \emph{Deformations of generalized calibrations and compact non-Kahler manifolds with vanishing first Chern class}, Asian Journal of Mathematics 7 (2003), 39-80 (\href{http://arxiv.org/abs/math/0205012}{arXiv:0205012})



\end{itemize}
[[!redirects calibrations]]

[[!redirects calibrated geometry]] [[!redirects calibrated geometries]]

[[!redirects calibrated manifold]] [[!redirects calibrated manifolds]]

[[!redirects calibrated submanifold]] [[!redirects calibrated submanifolds]]

[[!redirects manifold with calibration]] [[!redirects manifolds with calibration]] [[!redirects manifold with calibrations]] [[!redirects manifolds with calibrations]]



\end{document}