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% LaTeX version 2005/12/01 or later.
%
% This file contains the documentation of all differential geometry related macros .
%
% Macros have to be described by (delete the first %)
% \DescribeMacro{\macro}
% Description and usage of the macro.
%
% The description will appear in the usage
% part of the documentation. Use \subsubsection{} etc. for structuring.
%
% The implementation of the macros defined here has to be written in
% chairxmathDiffgeo.dtx
%\fi
%
%\subsubsection{General Macros in Differential Geometry} \label{sec:Doc_GeneralMacrosDiffGeo}
%
% \DescribeMacro{\Lie}
% Lie derivative |\Lie_X f|: $\Lie_X f$
%
% \DescribeMacro{\Schouten}
% Schouten bracket |\Schouten{X,Y}|: $\Schouten{X, Y}$.
%
% \DescribeMacro{\Forms}
% Differential forms |\Forms(M)|: $\Forms(M)$
%
% \DescribeMacro{\ZdR}
% DeRham cocycles |\ZdR(M, \mathbb{C})|: $\ZdR(M, \mathbb{C})$ \\
% Uses |operatorfont|.
%
% \DescribeMacro{\BdR}
% DeRham coboundaries |\BdR(M, \mathbb{C})|: $\BdR(M, \mathbb{C})$ \\
% Uses |operatorfont|.
%
% \DescribeMacro{\HdR}
% DeRham cohomology |\HdR(M, \mathbb{C})|: $\HdR(M, \mathbb{C})$ \\
% Uses |operatorfont|.
%
% \DescribeMacro{\Diffeo}
% Diffeomorphism group |\Diffeo(M)|: $\Diffeo(M)$ \\
% Uses |operatorfont|.
%
% \DescribeMacro{\Diffop}
% Differential operators |\Diffop(M)|: $\Diffop(M)$ \\
% Uses |operatorfont|.
%
% \DescribeMacro{\loc}
% To be used as an index |M_\loc|: $M_\loc$ \\
% Uses |scriptfont|.
%
% \DescribeMacro{\germ}
% Germs of functions |\germ_p(f)|: $\germ_p(f)$ \\
% Uses |operatorfont|.
%
% \DescribeMacro{\prol}
% Prolongation map |\prol(f)|: $\prol(f)$ \\
% Uses |operatorfont|.
%
% \DescribeMacro{\NRbracket}
% Nijenhuis-Richardson bracket |\NRbracket{a, b}|: $\NRbracket{a, b}$ \\
% Uses |scriptfont|.
%
% \DescribeMacro{\FNbracket}
% Fröhlicher-Nijenhuis bracket |\FNbracket{a, b}|: $\FNbracket{a, b}$ \\
% Uses |scriptfont|.
%
% \DescribeMacro{\Manifolds}
% The category of manifolds |\Manifolds|: $\Manifolds$ \\
% Uses |categorynamefont|
%
%\subsubsection{Lie Groups and Principal Fiber Bundles}
%
% \DescribeMacro{\lefttriv}
% Left trivialization |\lefttriv|: $\lefttriv$ \\
% Uses |operatorfont|.
%
% \DescribeMacro{\righttriv}
% Right trivialization |\righttriv|: $\righttriv$ \\
% Uses |operatorfont|.
%
% \DescribeMacro{\Gau}
% Gauge group |\Gau(P)|: $\Gau(P)$ \\
% Uses |operatorfont|.
%
% \DescribeMacro{\Conn}
% Connection one-forms |\Conn(P)|: $\Conn(P)$ \\
% Uses |operatorfont|.
%
% \DescribeMacro{\ratio}
% Ratio map of principal fiber bundle |\ratio(u, v)|: $\ratio(u, v)$ \\
% Uses |operatorfont|.
%
% \DescribeMacro{\Parallel}
% Parallel transport |\Parallel_{0 \to 1, \gamma}(v)|: $\Parallel_{0 \to 1, \gamma}(v)$ \\
% Uses |operatorfont|.
%
% \DescribeMacro{\CE}
% Chevalley-Eilenberg as index |C_\CE|: $C_\CE$ \\
% Uses |scriptfont|.
%
% \DescribeMacro{\HCE}
% Chevalley-Eilenberg cohomology |\HCE(\liealg{g})|: $\HCE(\liealg{g})$ \\
% Uses |operatorfont|.
%
% \DescribeMacro{\fund}
% Trivialization by fundamental vector fields |\fund|: $\fund$ \\
% Uses |operatorfont|.
%
% \DescribeMacro{\Universal}
% Universal enveloping algebra |\Universal{\liealg{g}}|: $\Universal(\liealg{g})$\\
% Uses |operatorfont|.
%
% \DescribeMacro{\BCH}
% BCH as small index |\sigma_\BCH|: $\sigma_\BCH$\\
% Uses |scriptfont|.
%
% \DescribeMacro{\LieGroups}
% The category of Lie groups |\LieGroups|: $\LieGroups$ \\
% Uses |categorynamefont|.
%
% \DescribeMacro{\Principal}
% The category of principal bundles |\Principal|: $\Principal$ \\
% Uses |categorynamefont|.
%
% \DescribeMacro{\GPrincipal}
% The category of $G$-principal bundles |\GPrincipal|: $\GPrincipal$ \\
% or with optional structure group |\GPrincipal[H]|: $\GPrincipal[H]$ \\
% Uses |categorynamefont|.
%
% \DescribeMacro{\Fiber}
% The category of fiber bundles |\Fiber|: $\Fiber$
% Uses |categorynamefont|.
%
% \DescribeMacro{\FFiber}
% The category of fiber bundles with typical fiber |\FFiber|:
% $\FFiber$ \\
% or with specified typical fiber |\FFiber[X]|: $\FFiber[X]$ \\
% Uses |categorynamefont|.
%
% \DescribeMacro{\Pin}
% The pin group |\Pin(q, p)|: $\Pin(p, q)$ \\
% Uses |groupfont|.
%
% \DescribeMacro{\Spin}
% The spin group |\Spin(q, p)|: $\Spin(p, q)$ \\
% Uses |groupfont|.
%
%\subsubsection{(Pseudo-) Riemannian Geometry}
%
% \DescribeMacro{\nablaLC}
% Levi-Civita covariant derivative |\nablaLC_X Y|: $\nablaLC_X Y$ \\
% Uses |scriptfont|.
%
% \DescribeMacro{\Laplace}
% Laplace operator |\Laplace f|: $\Laplace f$
%
% \DescribeMacro{\dAlembert}
% D'Alembert operator |\dAlembert u|: $\dAlembert u$
%
% \DescribeMacro{\feynman}
% Feynman slash notation |\feynman{D} = \feynman{A} + \feynman{\partial}|: 
% $\feynman{D} = \feynman{A} + \feynman{\partial}$
%
% \DescribeMacro{\Dirac}
% Dirac operator |\Dirac u|: $\Dirac u$
%
% \DescribeMacro{\rotation}
% Rotation (i.e. curl) of a vector field |\rotation(X)|: $\rotation(X)$. Not to be confused with $\textrm{grün}(X)$. \\
% Uses |operatorfont|.
%
% \DescribeMacro{\curl}
% Curl of a vector field |\curl \vec{X}|: $\curl \vec{X}$ \\
% Uses |operatorfont|.
%
% \DescribeMacro{\divergence}
% Divergence of a vector field |\divergence(X)|: $\divergence(X)$ \\
% Uses |operatorfont|.
%
% \DescribeMacro{\gradient}
% Gradient of a function |\gradient f|: $\gradient f$ \\
% Uses |operatorfont|.
%
% \DescribeMacro{\Tor}
% Torsion of a covariant derivative |\Tor (X, Y)|: $\Tor (X, Y)$ \\
% Uses |operatorfont|.
%
% \DescribeMacro{\Ric}
% Ricci curvature |\Ric (X, Y)|: $\Ric (X, Y)$ \\
% Uses |operatorfont|.
%
% \DescribeMacro{\scal}
% Scalar curvature |\scal|: $\scal$ \\
% Uses |operatorfont|.
%
% \DescribeMacro{\Riem}
% The set of Riemannian metrics (linear and on manifolds) |\Riem(M)|: $\Riem(M)$ \\
% Uses |operatorfont|.
%
% \DescribeMacro{\Hessian}
% Hessian of a function |\Hessian(f) \in \Secinfty(\Sym^2T^*M)|: $\Hessian(f) \in \Secinfty(\Sym^2T^*M)$ \\
% Uses |operatorfont|.
%
% \DescribeMacro{\hodge}
% Hodge star operator |\alpha \mapsto \hodge\alpha|: $\alpha \mapsto \hodge\alpha$
%
%\subsubsection{Complex Geometry}
%
% \DescribeMacro{\Nijenhuis}
% Nijenhuis operator |\Nijenhuis(X, Y)|: $\Nijenhuis(X, Y)$ \\
% Uses |operatorfont|.
%
% \DescribeMacro{\del}
% Dolbeault operator |\del \omega|: $\del \omega$
%
% \DescribeMacro{\delbar}
% CC of Dolbeault operator |\delbar\alpha|: $\delbar\alpha$
%
% \DescribeMacro{\FS}
% Fubini Study as very small index |\omega_\FS|: $\omega_\FS$ \\
% Uses |scriptfont|.
%
%\subsubsection{Vector Bundles}
%
% \DescribeMacro{\Lift}
% Generic lift of something |\nabla^\Lift|: $\nabla^\Lift$ \\
% Uses |scriptfont|.
%
% \DescribeMacro{\ver}
% Vertical lift |X^\ver|: $X^\ver$ \\
% Uses |scriptfont|.
%
% \DescribeMacro{\hor}
% Horizontal lift |X^\hor|: $X^\hor$ \\
% Uses |scriptfont|.
%
% \DescribeMacro{\Ver}
% Vertical subbundle |\Ver(E)|: $\Ver(E)$ \\
% Uses |operatorfont|.
%
% \DescribeMacro{\Hor}
% Horizontal subbundle |\Hor(E)|: $\Hor(E)$ \\
% Uses |operatorfont|.
%
% \DescribeMacro{\Sec}
% $C^k$-sections |\Sec(E)|: $\Sec(E)$ and |\Sec[2](E)|: $\Sec[2](E)$
%
% \DescribeMacro{\Secinfty}
% Smooth sections |\Secinfty(E)|: $\Secinfty(E)$
%
% \DescribeMacro{\HolSec}
% Holomorphic sections |\HolSec(U, E)|: $\HolSec(U, E)$ \\
% Uses |scriptfont|.
%
% \DescribeMacro{\SymD}
% Symmetrized covariant derivative |\SymD^n f|: $\SymD^n f$ \\
% Uses |operatorfont|.
%
% \DescribeMacro{\Densities}
% Densities of a vector bundle of rank $n$ or specific rank |\Densities TM|: $\Densities TM$
% and |\Densities[k]^\alpha E|: $\Densities[k]^\alpha E$.
%
% \DescribeMacro{\MeasurableSections}
% Measurable sections |\MeasurableSections(E)|: $\MeasurableSections(E)$ \\
% Uses |spacefont|.
%
% \DescribeMacro{\IntpSections}
% $p$-Integrable Sections |\IntpSections(\Densities T^*M)|: $\IntpSections(\Densities T^*M)$
% or with optional argument |\IntpSections[q](\Densities T^*M)|: $\IntpSections[q](\Densities T^*M)$.
%
% \DescribeMacro{\IntegrableSections}
% Integrable sections |\IntegrableSections(\Densities T^*M)|: $\IntegrableSections(\Densities T^*M)$
%
% \DescribeMacro{\Translation}
% Fiber translations |\Translation_A|: $\Translation_A$ \\
% Uses |operatorfont|.
%
% \DescribeMacro{\frames}
% Font for local frames |\frames{e}_1, \ldots, \frames{e}_k|: $\frames{e}_1, \ldots, \frames{e}_k$ \\
% Uses |operatorfont|.
%
% \DescribeMacro{\Frames}
% Frame bundle of a vector bundle |\Frames(E) \longrightarrow M|:\\
% $\Frames(E) \longrightarrow M$ \\
% Uses |operatorfont|.
%
%\DescribeMacro{\FDiff}
% Fiber derivative |\FDiff L|: $\FDiff L$ \\
% Uses |operatorfont|.
%
%\subsubsection{Symplectic and Poisson Geometry}
%
% \DescribeMacro{\Sympl}
% Symplectomorphism group |\Sympl(M, \omega)|: $\Sympl(M, \omega)$ \\
% Uses |groupfont|.
%
% \DescribeMacro{\Jacobiator}
% Jacobiator |\Jacobiator|: $\Jacobiator$
% and |\Jacobiator[\nu]|: $\Jacobiator[\nu]$ \\
% Uses |operatorfont|.
%
% \DescribeMacro{\red}
% Reduced as an index |M_\red|: $M_\red$ \\
% Uses |scriptfont|.
%
% \DescribeMacro{\Hess}
% Hess map |\Hess|: $\Hess(\nabla)$ \\
% Uses |operatorfont|.
%
% \DescribeMacro{\KKS}
% KKS as tiny index |\{f, g\}_\KKS|: $\{f, g\}_\KKS$ \\
% Uses |scriptfont|.
%
% \DescribeMacro{\Courant}
% Courant bracket |\Courant{a, b}|: $\Courant{a, b}$ \\
% Uses |scriptfont|.
%
% \DescribeMacro{\Dorfman}
% Dorfman bracket |\Dorfman{(x, \xi), (y, \eta)}|:
% $\Dorfman{(x, \xi), (y, \eta)}$ \\
% Uses |scriptfont|
%
% \DescribeMacro{\Dir}
% (Linear) Dirac structures |\Dir(V)|: $\Dir(V)$ \\
% Uses |operatorfont|.
%
% \DescribeMacro{\Forward}
% Forward map |\Forward(\phi)|: $\Forward(\phi)$ 
%
% \DescribeMacro{\Backward}
% Backward map |\Backward(\phi)|: $\Backward(\phi)$ 
%
% \DescribeMacro{\Tangent}
% Generalized tangent bundle/map |\Tangent M|: $\Tangent M$ 
%
% \DescribeMacro{\MWreduction}
% Marsden-Weinstein reduction |M \MWreduction G|: $M \MWreduction G$ 
%
% \DescribeMacro{\Mon}
% Monodromy groupoid |\Mon(M)|: $\Mon(M)$ \\
% Uses |operatorfont|.
%
% \DescribeMacro{\Hol}
% Holonomy groupoid |\Hol(M)|: $\Hol(M)$ \\
% Uses |operatorfont|.
%
% \endinput
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