Commit 0649df86 authored by Roland Herzog's avatar Roland Herzog

improves some symbols; adds commands for second covariant derivatives; improves comments

parent 7c030143
......@@ -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}{)}}%
......
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