\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*{lagrangian submanifold}

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

\hypertarget{symplectic_geometry}{}\paragraph*{{Symplectic geometry}}\label{symplectic_geometry}

[[!include symplectic geometry - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples_in_higher_differential_geometry}{Examples in higher differential geometry}\dotfill \pageref*{examples_in_higher_differential_geometry} \linebreak
\noindent\hyperlink{OfAPoissonLieAlgebroid}{Of a Poisson Lie algebroid}\dotfill \pageref*{OfAPoissonLieAlgebroid} \linebreak
\noindent\hyperlink{OfACourantLie2Algebroid}{Of a Courant Lie 2-algebroid}\dotfill \pageref*{OfACourantLie2Algebroid} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A \emph{Lagrangian submanifold} of a [[symplectic manifold]] is a submanifold which is a maximal [[isotropic submanifold]], hence a [[submanifold]] on which the [[symplectic form]] vanishes, and which is maximal with this property.

In the archetypical example of an even dimensional [[Cartesian space]] $X = \mathbb{R}^{2n}$ equipped with its canonical symplectic form $\omega = \sum_{i = 1}^n d q_i \wedge d p^i$, standard Lagrangian submanifolds are the submanifolds $\mathbb{R}^n \hookrightarrow \mathbb{R}^{2n}$ of fixed values of the $\{q_i\}_{i = 1}^n$ [[coordinates]]. Indeed \emph{locally}, every Lagrangian submanifold looks like this.

Lagrangian submanifolds are of central importance in [[symplectic geometry]] where they constitute [[leaves]] of [[real polarizations]] and are closely related to [[quantum states]]:

If one thinks of a [[symplectic manifold]] as a [[phase space]] of a [[physical system]], then a Lagrangian submanifold may be thought of (locally) as the space of ``all [[canonical momenta]] (= parameterization of a [[leaf]]) at fixed [[canonical coordinate]] (= parameterization of [[leaf space]])''.

A Lagrangian submanifold equipped with a [[half-density]] is a model for a [[state]] of the physical system in [[semiclassical approximation]] (see e.g. \hyperlink{BatesWeinstein}{Bates-Weinstein, p. 14}). A [[quantum state]] given by a [[wave function]] (see there) is a refinement of this concept.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\begin{defn}
\label{}\hypertarget{}{}
A (\textbf{Lagrangean} or) \textbf{lagrangian submanifold} of a [[symplectic manifold]] $(X,\omega)$ is a [[submanifold]] $L \hookrightarrow X$ such that the following equivalent conditions hold

\begin{itemize}%
\item at each point $\ell \in L the$[[tangent space]] $T_\ell L \hookrightarrow T_\ell X$ is a [[Lagrangian subspace]] (hence a simultanously [[isotropic subspace]] and [[cosisotropic subspace]]) of $T_\ell X$ equiiped with the [[symplectic form]] $\omega_x$;


\item $L \hookrightarrow X$ is a maximal [[isotropic submanifold]].



\end{itemize}
\end{defn}
\begin{remark}
\label{LambdaStructures}\hypertarget{LambdaStructures}{}
More generally one can consider Lagrangian submanifolds of symplectic structures in [[higher geometry]], such as [[symplectic Lie n-algebroids]] equipped with their canonical [[invariant polynomials]], thought of as [[dg-manifolds]] (via their [[Chevalley-Eilenberg algebra]]) and equipped with graded symplectic forms. Lagrangian dg-submanifolds of such symplectic dg-manifolds have been called \textbf{$\Lambda$-structures} in (\hyperlink{Severa}{\v{S}evera, section 4}).

\end{remark}
\hypertarget{examples_in_higher_differential_geometry}{}\subsection*{{Examples in higher differential geometry}}\label{examples_in_higher_differential_geometry}

We discuss classes of examples of Lagrangian dg-submanifolds, remark \ref{LambdaStructures}, of [[symplectic Lie n-algebroids]].

\hypertarget{OfAPoissonLieAlgebroid}{}\subsubsection*{{Of a Poisson Lie algebroid}}\label{OfAPoissonLieAlgebroid}

A [[Poisson Lie algebroid]] $\mathfrak{P}$ is a [[symplectic Lie n-algebroid]] for $n = 1$. Regarding its [[Chevalley-Eilenberg algebra]] as the algebra of functions on a [[dg-manifold]], that dg-manifold carries a graded [[symplectic form]] $\omega$. One can then say

\begin{defn}
\label{ForPoissonLieAlgebroidyByLagrangianFoliation}\hypertarget{ForPoissonLieAlgebroidyByLagrangianFoliation}{}
A dg-[[Lagrangian submanifold]] of $(\mathfrak{P}, \omega)$ is a Lagrangian dg-submanifold, also called a \textbf{$\Lambda$-structure}. (\hyperlink{Severa}{\v{S}evera, section 4}).

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
A [[foliation]] by such [[leaves]] is a Lagrangian \emph{[[foliation of a Lie algebroid]]}.

\end{remark}
\begin{prop}
\label{}\hypertarget{}{}
For $(X, \pi)$ the [[Poisson manifold]] underlying a [[Poisson Lie algebroid]] $(\mathfrak{P}, \omega)$, a dg-Lagrangian submanifold of $(\mathfrak{P}, \omega)$ corresponds to a [[coisotropic submanifold]] of $(X, \pi)$.

\end{prop}
(\hyperlink{Severa}{\v{S}evera, section 4})

\begin{proof}
As a [[vector bundle]] with bracket structure, the [[Poisson Lie algebroid]] $\mathfrak{P}$ is

\begin{displaymath}
\itexarray{
    T^* X &&\stackrel{\pi}{\to}&& T X
    \\
    & \searrow && \swarrow
    \\
    && X
  }
\end{displaymath}
where the horizontal morphism is given by contraction/pairing with the [[Poisson tensor]].

It is sufficient to consider this locally over a [[coordinate chart]] and hence we set without essential restriction of generality $X = \mathbb{R}^n$ with the [[invariant polynomial]]/graded [[symplectic form]] on $CE(\mathfrak{P})$ being

\begin{displaymath}
\omega = \mathbf{d} x^i \wedge \mathbf{d} p_i
  \,,
\end{displaymath}
where the $\{q_i\}_{i = 1}^n$ are the canonical [[coordinates]] on $\mathbb{R}^n$ and where the $\{p_i\}$ are the canonical coordinates on $T^*_x \mathbb{R}^n \simeq \mathbb{R}^n$, regarded as being in degree 1.

Consider then a sub-Lie algebroid of $\mathfrak{P}$ over a [[submanifold]] $S \hookrightarrow \mathbb{R}^n$. That the corresponding subbundle

\begin{displaymath}
\itexarray{
    E &\hookrightarrow& T^* X
    \\
    \downarrow && \downarrow
    \\
    S &\hookrightarrow & X
  }
\end{displaymath}
over $S$ is [[Lagrangian subspace|Lagrangian]] with respect to the above $\omega$ means that $E$ consists of precisely those [[cotangent vectors]] to $X$ which vanish when evaluated on [[tangent vectors]] of $S$. Hence

\begin{displaymath}
E = N^* S
\end{displaymath}
is the [[conormal bundle]] to $S \hookrightarrow X$. The inclusion $N^* S \hookrightarrow T^*_S X$ of vector bundles is an inclusion of [[Lie algebroids]] over $S$ precisely if the [[anchor map]] restricts to an anchor on $S$, hence that contraction with the Poisson tensor restricted to conormal vectors of $S$ lands in tangent vectors of $S$:

\begin{displaymath}
\pi(N^* S) \subset T S
  \,.
\end{displaymath}
This is the standard definition for what it means for $S$ to be a [[coisotropic submanifold]].

\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
The dg-Lagrangian submanifolds also correspond to [[branes]] in the [[Poisson sigma-model]] (see there) on $(\mathfrak{P}, \omega)$.

\end{remark}
\hypertarget{OfACourantLie2Algebroid}{}\subsubsection*{{Of a Courant Lie 2-algebroid}}\label{OfACourantLie2Algebroid}

A [[Courant Lie algebroid]] $\mathfrak{C}$ is a [[symplectic Lie n-algebroid]] for $n = 2$. Regarding its [[Chevalley-Eilenberg algebra]] as the algebra of functions on a [[dg-manifold]], that dg-manifold carries a graded [[symplectic form]] $\omega$. One can then say

\begin{defn}
\label{ForPoissonLieAlgebroidyByLagrangianFoliation}\hypertarget{ForPoissonLieAlgebroidyByLagrangianFoliation}{}
A dg-[[Lagrangian submanifold]] of $(\mathfrak{C}, \omega)$ is also called a \textbf{$\Lambda$-structure}. (\hyperlink{Severa}{\v{S}evera, section 4}).

Hence we might say \textbf{real polarization} of $(\mathfrak{C}, \omega)$ is a foliation by dg-Lagrangian submanifolds.

\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
The dg-Lagrangian submanifolds of a Courant Lie 2-algebroid $(\mathfrak{C}, \omega)$ correspond to [[Dirac structures]] on $(\mathfrak{C}, \omega)$.

\end{prop}
(\hyperlink{Severa}{\v{S}evera, section 4})

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

\begin{itemize}%
\item [[special Lagrangian submanifold]]


\item [[Lagrangian subspace]], [[Lagrangian Grassmannian]]


\item [[Lagrangian correspondence]], [[prequantized Lagrangian correspondence]]


\item [[Lagrangian cobordism]]


\item [[Lagrangian subspace]]


\item [[isotropic submanifold]]


\item [[polarization]]


\item [[Legendrean submanifold]]



\end{itemize}
[[!include (co)isotropic subspaces - table]]

[[!include infinity-CS theory for binary non-degenerate invariant polynomial - table]]

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

The concept of lagrangian submanifold has been defined/named in

\begin{itemize}%
\item [[Victor Maslov]], \emph{Perturbation Theory and Asymptotic Methods} (MSU Publ., Moscow, 1965; English translation: Mir, Moscow, 1965).

\end{itemize}
An introduction with an eye towards [[geometric quantization]] is for instance in

\begin{itemize}%
\item Sean Bates, [[Alan Weinstein]], \emph{Lectures on the geometry of quantization}, \href{http://www.math.berkeley.edu/~alanw/GofQ.pdf}{pdf}

\end{itemize}
(pages 10 and onward and then section 4.3).

Lagrangian submanfolds of symplectic [[dg-manifolds]] are called ``$\Lambda$-structures'' in

\begin{itemize}%
\item [[Pavol Ševera]], \emph{Some title containing the words ``homotopy'' and ``symplectic'', e.g. this one} (\href{http://arxiv.org/abs/math/0105080}{arXiv:0105080})

\end{itemize}
[[!redirects lagrangian submanifold]] [[!redirects lagrangian submanifolds]] [[!redirects Lagrangian submanifold]] [[!redirects Lagrangian submanifolds]] [[!redirects lagrangean submanifold]] [[!redirects lagrangean submanifolds]] [[!redirects Lagrangean submanifold]] [[!redirects Lagrangean submanifolds]]

[[!redirects lagrangian manifold]] [[!redirects lagrangian manifolds]] [[!redirects Lagrangian manifold]] [[!redirects Lagrangian manifolds]] [[!redirects lagrangean manifold]] [[!redirects lagrangean manifolds]] [[!redirects Lagrangean manifold]] [[!redirects Lagrangean manifolds]]

[[!redirects Lagrangian dg-submanifold]] [[!redirects Lagrangian dg-submanifolds]] [[!redirects dg-Lagrangian submanifold]] [[!redirects dg-Lagrangian submanifolds]]



\end{document}