adds a few conversion, where the embedding is not isometric.

src/manifolds/CenteredMatrices.jl

@@ -101,7 +101,7 @@ where $c_i = \frac{1}{m}\sum_{j=1}^m p_{j,i}$ for $i = 1, \dots, n$.
 """
 project(::CenteredMatrices, ::Any)
 
-project!(M::CenteredMatrices, q, p) = copyto!(q, p .- mean(p, dims=1))
+project!(::CenteredMatrices, q, p) = copyto!(q, p .- mean(p, dims=1))
 
 @doc raw"""
     project(M::CenteredMatrices, p, X)
@@ -119,7 +119,7 @@ where $c_i = \frac{1}{m}\sum_{j=1}^m x_{j,i}$  for $i = 1, \dots, n$.
 """
 project(::CenteredMatrices, ::Any, ::Any)
 
-project!(M::CenteredMatrices, Y, p, X) = (Y .= X .- mean(X, dims=1))
+project!(::CenteredMatrices, Y, p::Any, X) = (Y .= X .- mean(X, dims=1))
 
 @generated representation_size(::CenteredMatrices{m,n,𝔽}) where {m,n,𝔽} = (m, n)
 

src/manifolds/GeneralizedGrassmann.jl

@@ -57,6 +57,28 @@ function GeneralizedGrassmann(
     return GeneralizedGrassmann{n,k,field,typeof(B)}(B)
 end
 
+@doc raw"""
+    change_metric(M::GeneralizedGrassmann, ::EuclideanMetric, p X)
+
+Change `X` to the corresponding representer of the gradient with respect to the scaled metric
+of the [`GeneralizedGrassmann`](@ref) `M`, i.e. `B\X`.
+"""
+function change_gradient(M::GeneralizedGrassmann, ::EuclideanMetric, p, X)
+    return M.B \ X
+end
+
+@doc raw"""
+    change_metric(M::GeneralizedGrassmann, ::EuclideanMetric, p X)
+
+Change `X` to the corresponding vector with respect to the metric of the [`GeneralizedGrassmann`](@ref) `M`,
+i.e. let ``B=LL'`` be the cholesky decomposition of the metrix `B` then the corresponding vector is ``L\X``.
+
+"""
+function change_metric(M::GeneralizedGrassmann, ::EuclideanMetric, p, X)
+    C2 = cholesky(M.B).L
+    return C2 \ X
+end
+
 @doc raw"""
     check_point(M::GeneralizedGrassmann{n,k,𝔽}, p)
 

src/manifolds/HyperbolicHyperboloid.jl

@@ -1,3 +1,16 @@
+@doc raw"""
+    change_gradient(M::Hyperbolic, ::EuclideanMetric, p, X)
+
+to change a Euclidena gradient from the embedding that was already projected onto the tangent space at `X`
+we only have to correct for the metric, which means, that the sign of the last entry changes.
+"""
+function change_gradient(::Hyperbolic, ::EuclideanMetric, p, X)
+    Y = copy(M, p, X)
+    Y[end] *= -1
+    return Y
+end
+
+# a metric change does not seem possible here
 
 function check_point(M::Hyperbolic, p; kwargs...)
     mpv = invoke(check_point, Tuple{supertype(typeof(M)),typeof(p)}, M, p; kwargs...)
@@ -325,7 +338,7 @@ component such that for the
 resulting `p` we have that its [`minkowski_metric`](@ref) is $⟨p,X⟩_{\mathrm{M}} = 0$,
 i.e. $X_{n+1} = \frac{⟨\tilde p, Y⟩}{p_{n+1}}$, where $\tilde p = (p_1,\ldots,p_n)$.
 """
-_hyperbolize(M::Hyperbolic, p, Y) = vcat(Y, dot(p[1:(end - 1)], Y) / p[end])
+_hyperbolize(::Hyperbolic, p, Y) = vcat(Y, dot(p[1:(end - 1)], Y) / p[end])
 
 @doc raw"""
     inner(M::Hyperbolic{n}, p, X, Y)

src/manifolds/HyperbolicPoincareBall.jl

@@ -1,3 +1,39 @@
+@doc raw"""
+    change_gradient(M::Hyperbolic{n}, ::EuclideanMetric, p::PoincareBallPoint, X::PoincareBallTVector)
+
+Since in the metric we always have the term `` α = \frac{2}{1-\sum_{i=1}^n p_i^2}`` per element,
+the correction for the gradient reads `` Z = \frac{1}{α^2}X``.
+"""
+function change_gradient(
+    ::Hyperbolic,
+    ::EuclideanMetric,
+    p::PoincareBallPoint,
+    X::PoincareBallTVector,
+)
+    α = 2 / (1 - norm(p.value)^2)
+    Y = copy(M, p, X)
+    Y.value ./= α^2
+    return Y
+end
+
+@doc raw"""
+    change_metric(M::Hyperbolic{n}, ::EuclideanMetric, p::PoincareBallPoint, X::PoincareBallTVector)
+
+Since in the metric we always have the term `` α = \frac{2}{1-\sum_{i=1}^n p_i^2}`` per element,
+the correction for the metric reads `` Z = \frac{1}{α}X``.
+"""
+function change_metric(
+    ::Hyperbolic,
+    ::EuclideanMetric,
+    p::PoincareBallPoint,
+    X::PoincareBallTVector,
+)
+    α = 2 / (1 - norm(p.value)^2)
+    Y = copy(M, p, X)
+    Y.value ./= α
+    return Y
+end
+
 function check_point(M::Hyperbolic{N}, p::PoincareBallPoint; kwargs...) where {N}
     mpv = check_point(Euclidean(N), p.value; kwargs...)
     mpv === nothing || return mpv

src/manifolds/PositiveNumbers.jl

@@ -53,6 +53,23 @@ function check_point(M::PositiveNumbers, p; kwargs...)
     return nothing
 end
 
+@doc raw"""
+    change_metric(M::PositiveNumbers, E::EuclideanMetric, p, X)
+
+Given a tangent vector ``X ∈ T_p\mathcal M``representing a gradient with respect to the [`EuclideanMetric`](@ref) `g_E`,
+this function changes into the positivity metric representation of
+[`PositiveNumbers`](lref) `M` for the same gradient.
+"""
+change_gradient(::PositiveNumbers, ::EuclideanMetric, p, X) = p .* X .* p
+
+@doc raw"""
+    change_metric(M::SymmetricPOsitiveDefinite, E::EuclideanMetric, p, X)
+
+Given a tangent vector ``X ∈ T_p\mathcal M`` with respect to the [`EuclideanMetric`](@ref) `g_E`,
+this function changes into the positivity metric of [`PositiveNumbers`](lref) `M` for the same gradient.
+"""
+change_metric(::PositiveNumbers, ::EuclideanMetric, p, X) = p .* X
+
 """
     check_vector(M::PositiveNumbers, p, X; kwargs...)