...

 ... ... @@ -9,6 +9,7 @@ \newcommand{\bitangentSpaceSymbol}{\TT^{**}} \newcommand{\cotangentSpaceSymbol}{\TT^*} \newcommand{\covariantDerivativeSymbol}{\textup{D}} \newcommand{\secondCovariantDerivativeSymbol}{\covariantDerivativeSymbol^2} \newcommand{\geodesicSymbol}{\gamma} \@ifundefined{widearc}{% \DeclareSymbolFont{yhlargesymbols}{OMX}{yhex}{m}{n} ... ... @@ -17,11 +18,11 @@ }{% \newcommand{\geodesicArcSymbol}{\widearc} } \newcommand{\parallelTransportSymbol}{\mathrm{P}} \newcommand{\retractionSymbol}{\mathrm{retr}} \newcommand{\parallelTransportSymbol}{\textup{P}} \newcommand{\retractionSymbol}{\textup{retr}} \newcommand{\tangentSpaceSymbol}{\TT} \newcommand{\tensorSpaceSymbol}{\TT} \newcommand{\vectorTransportSymbol}{\mathrm{T}} \newcommand{\vectorTransportSymbol}{\textup{T}} % Define a command for the bitangent space % Argument {#1} is the base point ... ... @@ -108,7 +109,7 @@ % Argument {#1} (optional) is used to scale the brackets around #4 % Argument {#2} is the source base point % Argument {#3} is the destination base point % Argument {#4} is a tangent vector from the source base points tangent space % Argument {#4} is a tangent vector from the source base point's tangent space \NewDocumentCommand{\parallelTransport}{ O{} m m d()}{% \parallelTransportSymbol_{#3\leftarrow#2}% \IfNoValueF{#4}{\enclose[#1]{(}{#4}{)}}% ... ... @@ -117,14 +118,14 @@ % Argument {#1} (optional) is used to scale the brackets around #4 % Argument {#2} is the source base point % Argument {#3} is the direction to transport into % Argument {#4} is a tangent vector from the source base points tangent space % Argument {#4} is a tangent vector from the source base point's tangent space \NewDocumentCommand{\parallelTransportDir}{ O{} m m d()}{% \parallelTransportSymbol_{#2,#3}% \IfNoValueF{#4}{\enclose[#1]{(}{#4}{)}}% } % Define a command for a retraction map % Argument {#1} is used to scale the brackets around #3 % Argument {#1} (optional) is used to scale the brackets around #3 % Argument {#2} is the source base point % Argument {#3} is a corresponding tangent vector \NewDocumentCommand{\retract}{ O{} m g }{% ... ... @@ -144,6 +145,12 @@ % The optional argument [#3] can be used to denote the base point or to specify the metric \NewDocumentCommand{\riemanniannorm}{ O{} m O{} }{\norm[#1]{#2}_{#3}} % Define a command for the second covariant derivative % Argument {#1} is the vector (field) which determines the first direction of differentiation % Argument {#2} is the vector (field) which determines the second direction of differentiation % The optimal argument (#3) denotes the tensor field being differentiated \NewDocumentCommand{\secondCovariantDerivative}{ m m O{} }{\secondCovariantDerivativeSymbol_{#1,#2}#3} % Define a command for the tangent space % Argument {#1} is the base point % The optional argument [#2] (default \MM) is the manifold ... ... @@ -169,7 +176,7 @@ % Argument {#1} (optional) is used to scale the brackets around #4 % Argument {#2} is the source base point % Argument {#3} is the destination base point % Argument {#4} (optional) is a tangent vector from the source base points tangent space % Argument {#4} (optional) is a tangent vector from the source base point's tangent space % Argument {#5} (optional) can be used to specify that this vector transport is from a retraction or symmetric \NewDocumentCommand{\vectorTransport}{ O{} m m d() O{}}{% \vectorTransportSymbol_{#3\leftarrow#2}^{#5}\IfNoValueF{#4}{\enclose[#1]{(}{#4}{)}}% ... ... @@ -179,7 +186,7 @@ % Argument {#1} (optional) is used to scale the brackets around #4 % Argument {#2} is the source base point % Argument {#3} is the dirction to transport to (using the retraction in #5) % Argument {#4} (optional) is a tangent vector from the source base points tangent space % Argument {#4} (optional) is a tangent vector from the source base point's tangent space % Argument {#5} (optional) can be used to specify that this vector transport is from a retraction or symmetric \NewDocumentCommand{\vectorTransportDir}{ O{} m m d() O{}}{% \vectorTransportSymbol_{#2,#3}^{#5}\IfNoValueF{#4}{\enclose[#1]{(}{#4}{)}}% ... ...