Apply suggestions from code review

Co-authored-by: Mateusz Baran <mateuszbaran89@gmail.com>

src/manifolds/Euclidean.jl

@@ -716,7 +716,7 @@ end
 Compute the Weingarten map ``\mathcal W_p`` at `p` on the [`Euclidean`](@ref) `M` with respect to the
 tangent vector ``X \in T_p\mathcal M`` and the normal vector ``V \in N_p\mathcal M``.
 
-Since this a flat space by itself, the result is always the zero tangent vector
+Since this a flat space by itself, the result is always the zero tangent vector.
 """
 Weingarten(::Euclidean, p, X, V)
 

src/manifolds/FixedRankMatrices.jl

@@ -569,9 +569,9 @@ riemannian_Hessian(M::FixedRankMatrices, p, G, H, X)
 
 function riemannian_Hessian!(M::FixedRankMatrices, Y, p, G, H, X)
     project!(M, Y, p, H)
-    T1 = (G * X.Vt) / diagm(p.S)
+    T1 = (G * X.Vt) / Diagonal(p.S)
     Y.U .+= T1 - p.U * (p.U' * T1)
-    T2 = (G' * X.U) / diagm(p.S)
+    T2 = (G' * X.U) / Diagonal(p.S)
     Y.Vt .+= (T2 - p.Vt' * (p.Vt * T2))
     return Y
 end

src/manifolds/PowerManifold.jl

@@ -325,7 +325,7 @@ end
     Y = Weingarten(M::AbstractPowerManifold, p, X, V)
     Weingarten!(M::AbstractPowerManifold, Y, p, X, V)
 
-Since the metric decouples, also the computation of the weingarten map
+Since the metric decouples, also the computation of the Weingarten map
 ``\mathcal W_p`` can be computed elementwise on the single elements of the [`PowerManifold`](https://juliamanifolds.github.io/ManifoldsBase.jl/stable/manifolds/#sec-power-manifold) `M`.
 """
 Weingarten(::AbstractPowerManifold, p, X, V)

src/manifolds/ProductManifold.jl

@@ -1668,7 +1668,7 @@ end
     Y = Weingarten(M::ProductManifold, p, X, V)
     Weingarten!(M::ProductManifold, Y, p, X, V)
 
-Since the metric decouples, also the computation of the weingarten map
+Since the metric decouples, also the computation of the Weingarten map
 ``\mathcal W_p`` can be computed elementwise on the single elements of the [`ProductManifold`](@ref) `M`.
 """
 Weingarten(::ProductManifold, p, X, V)

src/manifolds/Rotations.jl

@@ -386,7 +386,7 @@ function Base.show(io::IO, ::Rotations{n}) where {n}
 end
 
 @doc raw"""
-riemannian_Hessian(M::Rotations, p, G, H, X)
+    riemannian_Hessian(M::Rotations, p, G, H, X)
 
 The Riemannian Hessian can be computed by adopting Eq. (5.6) [Nguyen:2023](@cite),
 so very similar to the Stiefel manifold.

src/manifolds/SkewHermitian.jl

@@ -260,7 +260,7 @@ end
 Compute the Weingarten map ``\mathcal W_p`` at `p` on the [`SkewSymmetricMatrices`](@ref) `M` with respect to the
 tangent vector ``X \in T_p\mathcal M`` and the normal vector ``V \in N_p\mathcal M``.
 
-Since this a flat space by itself, the result is always the zero tangent vector
+Since this a flat space by itself, the result is always the zero tangent vector.
 """
 Weingarten(::SkewSymmetricMatrices, p, X, V)
 

src/manifolds/Sphere.jl

@@ -529,7 +529,7 @@ The formula is due to [AbsilMahonyTrumpf:2013](@cite) given by
 Weingarten(::Sphere, p, X, V)
 
 function Weingarten!(::Sphere, Y, p, X, V)
-    Y .= -X * (p'V)
+    Y .= -dot(p, V) .* X
     return Y
 end
 

src/manifolds/Symmetric.jl

@@ -241,7 +241,7 @@ end
 Compute the Weingarten map ``\mathcal W_p`` at `p` on the [`SymmetricMatrices`](@ref) `M` with respect to the
 tangent vector ``X \in T_p\mathcal M`` and the normal vector ``V \in N_p\mathcal M``.
 
-Since this a flat space by itself, the result is always the zero tangent vector
+Since this a flat space by itself, the result is always the zero tangent vector.
 """
 Weingarten(::SymmetricMatrices, p, X, V)
 

src/manifolds/Unitary.jl

@@ -117,7 +117,7 @@ show(io::IO, ::UnitaryMatrices{n,ℂ}) where {n} = print(io, "UnitaryMatrices($(
 show(io::IO, ::UnitaryMatrices{n,ℍ}) where {n} = print(io, "UnitaryMatrices($(n), ℍ)")
 
 @doc raw"""
-riemannian_Hessian(M::UnitaryMatrices, p, G, H, X)
+    riemannian_Hessian(M::UnitaryMatrices, p, G, H, X)
 
 The Riemannian Hessian can be computed by adopting Eq. (5.6) [Nguyen:2023](@cite),
 so very similar to the complex Stiefel manifold.

src/manifolds/VectorBundle.jl

@@ -1264,7 +1264,7 @@ end
 Compute the Weingarten map ``\mathcal W_p`` at `p` on the [`VectorSpaceAtPoint`](@ref) `M` with respect to the
 tangent vector ``X \in T_p\mathcal M`` and the normal vector ``V \in N_p\mathcal M``.
 
-Since this a flat space by itself, the result is always the zero tangent vector
+Since this a flat space by itself, the result is always the zero tangent vector.
 """
 Weingarten(::VectorSpaceAtPoint, p, X, V)