Improve docs of group exp and log, mention left-invariance

src/Lie_algebra/Lie_algebra_interface.jl

@@ -12,6 +12,20 @@ The Lie algebras considered here are those related to a [`LieGroup`](@ref) ``$(_
 namely the tangent space ``T_{$(_math(:e))}$(_math(:G))`` at the [`Identity`](@ref),
 this is internally just a `const` of the corresponding $(_link(:TangentSpace)).
 
+!!! note "Convention"
+    A vector field ``$(_tex(:Cal,"X")): $(_math(:G)) → T$(_math(:G))``, ``X(g) ∈ T_g$(_math(:G))``
+    is called a left-invariant vector field if it satisfies
+
+    ```math
+    $(_tex(:Cal,"X"))(λ_g(h)) = Dλ_g(h)[$(_tex(:Cal,"X"))(h)], $(_tex(:quad))$(_tex(:text, "for all"))$(_tex(:quad)) g, h ∈ $(_math(:G)),
+    ```
+
+    where ``λ_g: $(_math(:G)) → $(_math(:G))`` is the left multiplication by ``g``.
+    Hence ``$(_tex(:Cal,"X"))`` is determined already when ``X ∈ $(_math(:𝔤))`` is given,
+    since ``$(_tex(:Cal,"X"))(g) = Dλ_g(e)[X]``, cf [HilgertNeeb:2012; Definition 9.1.7](@cite).
+
+    Throughout `LieGrous.jl`, we use this left-invariant convention.
+
 # Constructor
 
     LieAlgebra(G::LieGroup)

src/documentation_glossary.jl

@@ -86,6 +86,8 @@ define!(
 )
 define!(:LaTeX, :sum, raw"\sum")
 define!(:LaTeX, :transp, raw"\mathrm{T}")
+define!(:LaTeX, :text, (text) -> raw"\text" * "{$text}")
+
 #
 # ---
 # Mathematics and semantic symbols

src/interface.jl

@@ -346,7 +346,9 @@ _doc_exp = """
     exp(G::LieGroup, g, X, t::Number=1)
     exp!(G::LieGroup, h, g, X, t::Number=1)
 
-Compute the Lie group exponential map given by
+Compute the Lie group exponential map for ``g∈$(_math(:G))`` and ``X∈$(_math(:𝔤))``,
+where ``$(_math(:𝔤))`` denotes the [`LieAlgebra`](@ref) of ``$(_math(:G))``.
+It is given by
 
 ```math
 $(_tex(:exp))_g X = g$(_math(:∘))$(_tex(:exp))_{$(_math(:G))}(X)
@@ -355,9 +357,8 @@ $(_tex(:exp))_g X = g$(_math(:∘))$(_tex(:exp))_{$(_math(:G))}(X)
 where `X` can be scaled by `t`, the computation can be performed in-place of `h`,
 and ``$(_tex(:exp))_{$(_math(:G))}`` denotes the  [Lie group exponential function](@ref exp(::LieGroup, ::Identity, :Any)).
 
-!!! note
-    If `g` is the [`Identity`](@ref) the [Lie group exponential function](@ref exp(::LieGroup, ::Identity, :Any)) is computed directly.
-    Implementing the Lie group exponential function introduces a default implementation for this function.
+If `g` is the [`Identity`](@ref) the [Lie group exponential function](@ref exp(::LieGroup, ::Identity, :Any)) ``$(_tex(:exp))_{$(_math(:G))}`` is computed directly.
+Implementing the Lie group exponential function introduces a default implementation with the formula above.
 
 !!! note
     The Lie group exponential map is usually different from the exponential map with respect
@@ -666,7 +667,7 @@ _doc_is_vector = """
 
 Check whether `X` is a tangent vector, that is an element of the [`LieAlgebra`](@ref)
 of `G`.
-The first variant calls [`is_point`](@extref ManifoldsBase.is_point) on the [`LieAlgebra`](@ref) `𝔤` of `G`.
+The first variant calls [`is_point`](@extref `ManifoldsBase.is_point-Tuple{AbstractManifold, Any, Bool}`) on the [`LieAlgebra`](@ref) `𝔤` of `G`.
 The second variant calls [`is_vector`](@extref ManifoldsBase.is_vector) on the $(_link(:AbstractManifold)) at the [`identity_element`](@ref).
 
 All keyword arguments are passed on to the corresponding call
@@ -717,7 +718,9 @@ _doc_log = """
     log(G::LieGroup, g, h)
     log!(G::LieGroup, X, g, h)
 
-Compute the Lie group logarithmic map
+Compute the Lie group logarithmic map ``$(_tex(:log))_g: $(_math(:G)) → $(_math(:𝔤))``,
+where ``$(_math(:𝔤))`` denotes the [`LieAlgebra`](@ref) of ``$(_math(:G))``.
+It is given by
 
 ```math
 $(_tex(:log))_g h = $(_tex(:log))_{$(_math(:G))}(g^{-1}$(_math(:∘))h)
@@ -726,9 +729,8 @@ $(_tex(:log))_g h = $(_tex(:log))_{$(_math(:G))}(g^{-1}$(_math(:∘))h)
 where ``$(_tex(:log))_{$(_math(:G))}`` denotes the [Lie group logarithmic function](@ref log(::LieGroup, ::Identity, :Any))
 The computation can be performed in-place of `X`.
 
-!!! note
-    If `g` is the [`Identity`](@ref) the [Lie group logarithmic function](@ref log(::LieGroup, ::Identity, :Any)) is computed directly.
-    Implementing the Lie group logarithmic function introduces a default implementation for this function.
+If `g` is the [`Identity`](@ref) the [Lie group logarithmic function](@ref log(::LieGroup, ::Identity, :Any)) ``$(_tex(:log))_{$(_math(:G))}`` is computed directly.
+Implementing the Lie group logarithmic function introduces a default implementation for this function with the formula above.
 
 !!! note
     The Lie group logarithmic map is usually different from the logarithmic map with respect