diff --git a/src/manifolds/Sphere.jl b/src/manifolds/Sphere.jl
index 9d6f6ba2e1..703b8cd924 100644
--- a/src/manifolds/Sphere.jl
+++ b/src/manifolds/Sphere.jl
@@ -534,7 +534,7 @@ 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 .= -X * (p'V)
     return Y
 end
 
diff --git a/src/manifolds/StiefelCanonicalMetric.jl b/src/manifolds/StiefelCanonicalMetric.jl
index b7a18f64c7..8c1d2da219 100644
--- a/src/manifolds/StiefelCanonicalMetric.jl
+++ b/src/manifolds/StiefelCanonicalMetric.jl
@@ -188,14 +188,14 @@ function inverse_retract!(
 end
 
 @doc raw"""
-    Y = riemannian_Hessian(M::MetricManifold{ℝ, Stiefel{n,k,ℝ}, CanonicalMetric}, p, G, H, X)
-    riemannian_Hessian!(M::MetricManifold{ℝ, Stiefel{n,k,ℝ}, CanonicalMetric}, Y, p, G, H, X)
+    Y = riemannian_Hessian(M::MetricManifold{ℝ, Stiefel{n,k}, CanonicalMetric}, p, G, H, X)
+    riemannian_Hessian!(M::MetricManifold{ℝ, Stiefel{n,k}, CanonicalMetric}, Y, p, G, H, X)
 
 Compute the Riemannian Hessian ``\operatorname{Hess} f(p)[X]`` given the
 Euclidean gradient ``∇ f(\tilde p)`` in `G` and the Euclidean Hessian ``∇^2 f(\tilde p)[\tilde X]`` in `H`,
 where ``\tilde p, \tilde X`` are the representations of ``p,X`` in the embedding,.
 
-Here, we adopt Eq. (5.6) [Nguyen:2023](@cite), for the [`CanonicalMetric`](https://juliamanifolds.github.io/ManifoldsBase.jl/stable/manifolds.html#ManifoldsBase.EuclideanMetric)
+Here, we adopt Eq. (5.6) [Nguyen:2023](@cite), for the [`CanonicalMetric`](@ref)
 ``α_0=1, α_1=\frac{1}{2}`` in their formula. The formula reads
 
 ```math
@@ -211,21 +211,21 @@ where ``P = I-pp^{\mathrm{H}}``.
 Compared to Eq. (5.6) we have ``α_0 = 1`` and ``α_1 = \frac{1}{2}``.
 """
 riemannian_Hessian(
-    M::MetricManifold{ℝ,Stiefel{n,k,ℝ},CanonicalMetric},
+    M::MetricManifold{𝔽,Stiefel{n,k,𝔽},CanonicalMetric},
     p,
     G,
     H,
     X,
-) where {n,k}
+) where {n,k,𝔽}
 
 function riemannian_Hessian!(
-    ::MetricManifold{ℝ,Stiefel{n,k,ℝ},CanonicalMetric},
+    M::MetricManifold{𝔽,Stiefel{n,k,𝔽},CanonicalMetric},
     Y,
     p,
     G,
     H,
     X,
-) where {n,k}
+) where {n,k,𝔽}
     project!(
         M,
         Y,
diff --git a/test/manifolds/sphere.jl b/test/manifolds/sphere.jl
index 41fac3e55d..f217fe4901 100644
--- a/test/manifolds/sphere.jl
+++ b/test/manifolds/sphere.jl
@@ -264,4 +264,12 @@ using ManifoldsBase: TFVector
         @test dpX[2][1] == 1.0
         @test dpX[2][2].X == normalize(X)
     end
+    @testset "Weingarten" begin
+        M = Sphere(2)
+        p = [1.0, 0.0, 0.0]
+        X = [0.0, 2.0, 0.0]
+        V = [0.3, 0.0, 0.0]
+        Y = -X * (p'V)
+        @test Weingarten(M, p, X, V) == Y
+    end
 end
diff --git a/test/manifolds/stiefel.jl b/test/manifolds/stiefel.jl
index edd765dc5e..aed954efa0 100644
--- a/test/manifolds/stiefel.jl
+++ b/test/manifolds/stiefel.jl
@@ -331,6 +331,8 @@ include("../utils.jl")
         @test inner(M3, p, X, Y) == 0
         @test inner(M3, p, X, 2 * X + 3 * Y) == 2 * inner(M3, p, X, X)
         @test norm(M3, p, X) ≈ distance(M3, p, q)
+        Z = [0.0 0.0; 0.0 0.0; -1.0 -1.0]
+        @test riemannian_Hessian(M3, p, Y, Z, X) == [0.0 0.5; -0.5 0.0; -1.0 -1.0]
         # check on a higher dimensional manifold, that the iterations are actually used
         M4 = MetricManifold(Stiefel(10, 2), CanonicalMetric())
         p = Matrix{Float64}(I, 10, 2)
diff --git a/test/manifolds/vector_bundle.jl b/test/manifolds/vector_bundle.jl
index b8d8018fd2..61265c4a70 100644
--- a/test/manifolds/vector_bundle.jl
+++ b/test/manifolds/vector_bundle.jl
@@ -271,4 +271,13 @@ struct TestVectorSpaceType <: VectorSpaceType end
         @test is_flat(M)
         @test injectivity_radius(M) == Inf
     end
+
+    @testset "Weingarten Map" begin
+        p0 = [1.0, 0.0, 0.0]
+        M = TangentSpaceAtPoint(Sphere(2), p0)
+        p = [0.0, 1.0, 1.0]
+        X = [0.0, 1.0, 0.0]
+        N = [1.0, 0.0, 0.0]
+        @test Weingarten(M, p, X, V) == zero_vector(M, p)
+    end
 end