Add a proper working default.

src/differentiation/riemannian_diff.jl

@@ -67,12 +67,12 @@ In the tangent space itself, this backend then employs an (Euclidean)
 
 # Constructor
 
-    TangentDiffBackend()
+    TangentDiffBackend(diff_backend)
+
+where `diff_backend` is an [`AbstractDiffBackend`](@ref) to be used on the tangent space.
 
 With the keyword arguments
 
-* `diff_backend` an [`AbstractDiffBackend`](@ref) to be used on the tangent space
-  (by defaut [`FiniteDiffBackend`](@ref) is used)
 * `retraction` an [`AbstractRetractionMethod`](@ref) ([`ExponentialRetraction`]('ref) by default)
 * `inverse_retraction` an [`AbstractInverseRetractionMethod`](@ref) ([`LogarithmicInverseRetraction`]('ref) by default)
 * `basis` an [`AbstractBasis`](@ref) ([`DefaultOrthogonalBasis`]('ref) by default)
@@ -88,8 +88,7 @@ struct TangentDiffBackend{
     inverse_retraction::TIR
     basis::TB
 end
-function TangentDiffBackend(;
-    diff_backend::TAD=FiniteDiffBackend(),
+function TangentDiffBackend(diff_backend::TAD;
     retraction::TR = ExponentialRetraction(),
     inverse_retraction::TIR = LogarithmicInverseRetraction(),
     basis::TB = DefaultOrthonormalBasis()
@@ -198,7 +197,7 @@ globally default differentiation backend for calculating gradients.
 
 [`Manifolds.gradient(::AbstractManifold, ::Any, ::Any, ::AbstractRiemannianDiffBackend)`](@ref)
 """
-const _current_rgradient_backend = CurrentRiemannianDiffBackend(TangentDiffBackend())
+const _current_rgradient_backend = CurrentRiemannianDiffBackend(TangentDiffBackend(FiniteDifferencesBackend()))
 
 """
     _current_rdifferential_backend
@@ -210,7 +209,7 @@ globally default differentiation backend for calculating differentials.
 
 [`Manifolds.differential`](@ref)
 """
-const _current_rdifferential_backend = CurrentRiemannianDiffBackend(TangentDiffBackend())
+const _current_rdifferential_backend = CurrentRiemannianDiffBackend(TangentDiffBackend(FiniteDifferencesBackend()))
 
 """
     rgradient_backend() -> AbstractRiemannianDiffBackend
@@ -246,7 +245,7 @@ function rdifferential_backend!(backend::AbstractRiemannianDiffBackend)
     return backend
 end
 
-"""
+@doc raw"""
     RiemannianProjectionBackend <: AbstractRiemannianDiffBackend
 
 This backend computes the differentiation in the embedding, which is currently limited