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diff --git a/Master/texmf-dist/source/latex/nchairx/chairxmathDiffgeoDoc.dtx b/Master/texmf-dist/source/latex/nchairx/chairxmathDiffgeoDoc.dtx new file mode 100644 index 00000000000..922f1acaca3 --- /dev/null +++ b/Master/texmf-dist/source/latex/nchairx/chairxmathDiffgeoDoc.dtx @@ -0,0 +1,362 @@ +% \iffalse meta-comment +% +% Copyright (C) 2018 - 2021 by ChairX +% +% This file may be distributed and/or modified under the +% conditions of the LaTeX Project Public License, either +% version 1.3 of this license or (at your option) any later +% version. The latest version of this license is in: +% +% http://www.latex-project.org/lppl.txt +% +% and version 1.3 or later is part of all distributions of +% 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 +% Local Variables: +% mode: doctex +% TeX-master: "nchairx" +% End: |