<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">% This LaTeX package provides numerous commands relevant to typesetting
% material related to manifolds and differential geometry.
\ProvidesPackage{mathsemantics-manifolds}[2022/05/06]

% Resolve the dependencies of this package
\RequirePackage{mathsemantics-semantic}

% Define some symbols
\newcommand{\bitangentSpaceSymbol}{\cT^{**}}
\newcommand{\cotangentSpaceSymbol}{\cT^*}
\newcommand{\covariantDerivativeSymbol}{\textup{D}}
\newcommand{\secondCovariantDerivativeSymbol}{\covariantDerivativeSymbol^2}
\newcommand{\geodesicSymbol}{\gamma}
\@ifundefined{widearc}{%
	\DeclareSymbolFont{yhlargesymbols}{OMX}{yhex}{m}{n}
	\DeclareMathAccent{\wideparen}{\mathord}{yhlargesymbols}{"F3}
	\newcommand{\geodesicArcSymbol}{\wideparen}
	}{%
	\newcommand{\geodesicArcSymbol}{\widearc}
}
\newcommand{\parallelTransportSymbol}{\textup{P}}
\newcommand{\retractionSymbol}{\textup{retr}}
\newcommand{\tangentSpaceSymbol}{\cT}
\newcommand{\tensorSpaceSymbol}{\cT}
\newcommand{\vectorTransportSymbol}{\textup{T}}

% Define a command for the bitangent space
% Argument {#1} is the base point
% The optional argument [#2] (default \cM) is the manifold
\NewDocumentCommand{\bitangentSpace}{ m O{\cM} }{\bitangentSpaceSymbol_{#1}#2}

% Define a command for the cotangent space
% Argument {#1} is the base point
% The optional argument [#2] (default \cM) is the manifold
\NewDocumentCommand{\cotangentSpace}{ m O{\cM} }{\cotangentSpaceSymbol_{#1}\!#2}

% Define a command for the cotangent bundle
% The optional argument [#1] (default \cM) is the manifold
\NewDocumentCommand{\cotangentBundle}{ O{\cM} }{\cotangentSpaceSymbol\!#1}

% Define a command for the covariant derivative (connection)
% Argument {#1} is the vector (field) which determines the direction of differentiation
% The optional argument [#2] denotes the tensor field being differentiated
\NewDocumentCommand{\covariantDerivative}{ m O{} }{\covariantDerivativeSymbol_{#1}#2}

\DeclareMathOperator{\expOp}{exp}
% Define a command for the exponential map
% Argument {#1} is used to scale the brackets around #3
% Argument {#2} is the source base point
% Argument {#3} is a corresponding tangent vector
\NewDocumentCommand{\exponential}{ O{} m d() }{%
    \expOp_{#2}\IfNoValueF{#3}{\enclose[#1]{(}{#3}{)}}%
}

% Define a command for various versions of geodesic curves
% * The optional argument (#1) can be used to locally set the geodesic symbol.
%     note that globally setting it, you can also redefine `\geodesicSymbol'
% * The optional argument &lt;#2&gt; defines the type: [s]symbol, [l]ong, [a]rc, or [p]lain (the default)
% * The optional argument [#2] scales the delimiters
% * Argument {#4} is the initial point (t = 0) of the geodesic
% * Argument {#5} is the final point (t = 1) of the geodesic (for types l and a)
% * Argument {#5} is the initial tangent direction of the geodesic (for type p)
% * The optional argument (#6) denotes the evaluation point
\NewDocumentCommand{\geodesic}{ D||{\geodesicSymbol} D&lt;&gt;{plain} o G{} G{} d() }{%
	\ifthenelse{\equal{#2}{s}}{%
		{#1}% format: just the symbol
		\IfNoValueF{#6}{\enclose[#3]{(}{#6}{)}}% if arg given
		}{% not s
		\ifthenelse{\equal{#2}{l}}{% long form
			{#1}% format:  arg; p1, p2
			\enclose[#3]{(}{%
				\IfNoValueTF{#6}{\cdot}{#6};#4\,,\,#5%
			}{)}%print \cdot if not param given
			}{% neither s nor l
			\ifthenelse{\equal{#2}{a}}{%
				{#1}_{\geodesicArcSymbol{#4,#5}}%
				\IfNoValueF{#6}{\enclose[#3]{(}{#6}{)}}% if arg given -&gt; print
				}{% neither s nor l nor a -&gt; plain
				{#1}_{#4,#5}%
				\IfNoValueF{#6}{\enclose[#3]{(}{#6}{)}}% if arg given -&gt; print
			}%
		}%
	}%
}

\DeclareMathOperator{\logOp}{log}
% Define a command for the logarithmic map
% Argument {#1} is used to scale the brackets around #3
% Argument {#2} is the source base point
% Argument {#3} is a second point
\NewDocumentCommand{\logarithm}{ O{} m d() }{%
    \logOp_{#2}\IfNoValueF{#3}{\enclose[#1]{(}{#3}{)}}%
}

% Define a command for the logarithmic map
% Argument {#1} is used to scale the brackets around #3
% Argument {#2} is the source base point
% Argument {#3} is a second point
\NewDocumentCommand{\inverseRetract}{ O{} m d() }{%
    \retractionSymbol^{-1}_{#2}\IfNoValueF{#3}{\enclose[#1]{(}{#3}{)}}%
}

% Define a command for the Lie bracket (of two vector fields)
% The optional argument [#1] scales the delimiters
% Argument {#2} is the first vector field
% Argument {#3} is the second vector field
\NewDocumentCommand{\lie}{ O{} m m }{\enclose[#1]{[}{#2,#3}{]}}

% Define a command for the parallel transport, where
% 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 (in round brackets)
% Argument {#5} (optional) can be used to specify that this parallel transport is along a curve
\NewDocumentCommand{\parallelTransport}{ O{} m m d() O{}}{%
    \parallelTransportSymbol^{#5}_{#3\leftarrow#2}%
        \IfNoValueF{#4}{\enclose[#1]{(}{#4}{)}}%
}

% Define a command for the parallel transport with direction, where
% 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 (in round brackets)
% Argument {#5} (optional) can be used to specify that this parallel transport is along a curve
\NewDocumentCommand{\parallelTransportDir}{ O{} m m d() O{}}{%
    \parallelTransportSymbol^{#5}_{#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 {#2} is the source base point
% Argument {#3} is a corresponding tangent vector
\NewDocumentCommand{\retract}{ O{} m d() }{%
    \retractionSymbol_{#2}\IfNoValueF{#3}{\enclose[#1]{(}{#3}{)}}%
}

% Define a command for the Riemannian metric (inner product in the tangent space)
% The optional argument [#1] scales the delimiters
% Argument {#2} is the first tangent vector
% Argument {#3} is the second tangent vector
% The optional argument [#4] can be used to denote the base point or to specify the metric
\NewDocumentCommand{\riemannian}{ O{} m m O{} }{\enclose[#1]{(}{#2\,,\,#3}{)}_{#4}}

% Define a command for the norm induced by the Riemannian metric (inner product in the tangent space)
% The optional argument [#1] scales the delimiters
% Argument {#2} is the tangent vector
% 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 \cM) is the manifold
\NewDocumentCommand{\tangentSpace}{ m O{\cM} }{\tangentSpaceSymbol_{#1}#2}

% Define a command for the tangent bundle
% The optional argument [#1] (default \cM) is the manifold
\NewDocumentCommand{\tangentBundle}{ O{\cM} }{\tangentSpaceSymbol{}#1}

% Define a command for the (r,s)-tensor bundle
% Argument {#1} is the number r of elements of the cotangent space the tensors accept.
% Argument {#2} is the number s of elements of the tangent space the tensors accept.
% The optimal argument [#3] (default \cM) is the manifold
\NewDocumentCommand{\tensorBundle}{ m m O{\cM} }{\tensorSpaceSymbol^{(#1,#2)}#3}

% Define a command for the space of (r,s)-tensors over a vector space V
% Argument {#1} is the number r of elements of the dual space V^* the tensors accept.
% Argument {#2} is the number s of elements of the space V the tensors accept.
% The optimal argument [#3] (default empty) denotes the vector space V.
\NewDocumentCommand{\tensorSpace}{ m m O{} }{\tensorSpaceSymbol^{(#1,#2)}\IfNoValueF{#3}{(#3)}}

% Define a command for the general vector transport, where
% 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 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}{)}}%
}

% Define a command for the general vector transport with direction, where
% 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 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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