diff --git a/NEWS.md b/NEWS.md
index 1a614167..b457a004 100644
--- a/NEWS.md
+++ b/NEWS.md
@@ -5,11 +5,14 @@ All notable changes to this project will be documented in this file.
 The format is based on [Keep a Changelog](https://keepachangelog.com/en/1.0.0/),
 and this project adheres to [Semantic Versioning](https://semver.org/spec/v2.0.0.html).
 
-## [0.15.10] 15/05/2024
+## [0.15.10] 19/05/2024
 
 ### Added
 
 * Functions `fill(p, N)` and `fill!(P, p, N)` to fill values into a point on a power manifold `N`.
+* introduce a `base_point(TpM)` to access the base point of a tangent space
+* introduce `TpM[i]` to access tangent spaces of factors from an `AbstractPowerManifold` or a `ProductManifold`.
+
 
 ## [0.15.9] 02/05/2024
 
diff --git a/src/ManifoldsBase.jl b/src/ManifoldsBase.jl
index 3c946558..43b30f4a 100644
--- a/src/ManifoldsBase.jl
+++ b/src/ManifoldsBase.jl
@@ -1221,6 +1221,7 @@ export ×,
     allocate,
     angle,
     base_manifold,
+    base_point,
     change_basis,
     change_basis!,
     change_metric,
diff --git a/src/PowerManifold.jl b/src/PowerManifold.jl
index e75266a6..c188c713 100644
--- a/src/PowerManifold.jl
+++ b/src/PowerManifold.jl
@@ -820,6 +820,21 @@ Base.@propagate_inbounds function Base.getindex(
     return get_component(M, p, I...)
 end
 
+"""
+    getindex(M::TangentSpace{𝔽, AbstractPowerManifold}, i...)
+    TpM[i...]
+
+Access the `i`th manifold component from an [`AbstractPowerManifold`](@ref)s' tangent space `TpM`.
+"""
+function Base.getindex(
+    TpM::TangentSpace{𝔽,<:AbstractPowerManifold},
+    I::Union{Integer,Colon,AbstractVector}...,
+) where {𝔽}
+    M = base_manifold(TpM)
+    p = base_point(TpM)
+    return TangentSpace(M.manifold, p[M, I...])
+end
+
 @doc raw"""
     injectivity_radius(M::AbstractPowerManifold[, p])
 
diff --git a/src/ProductManifold.jl b/src/ProductManifold.jl
index f5b4b80c..3a4d42bd 100644
--- a/src/ProductManifold.jl
+++ b/src/ProductManifold.jl
@@ -28,6 +28,17 @@ access the `i`th manifold component from the [`ProductManifold`](@ref) `M`.
 """
 @inline Base.getindex(M::ProductManifold, i::Integer) = M.manifolds[i]
 
+"""
+    getindex(M::TangentSpace{𝔽,<:ProductManifold}, i::Integer)
+    TpM[i]
+
+Access the `i`th manifold component from a [`ProductManifold`](@ref)s' tangent space `TpM`.
+"""
+function Base.getindex(TpM::TangentSpace{𝔽,<:ProductManifold}, i::Integer) where {𝔽}
+    M = base_manifold(TpM)
+    return TangentSpace(M[i], base_point(TpM)[M, i])
+end
+
 ProductManifold() = throw(MethodError("No method matching ProductManifold()."))
 
 const PRODUCT_BASIS_LIST = [
diff --git a/src/TangentSpace.jl b/src/TangentSpace.jl
index 8ba795ef..91a7a88e 100644
--- a/src/TangentSpace.jl
+++ b/src/TangentSpace.jl
@@ -40,6 +40,13 @@ function allocate_result(M::TangentSpace, ::typeof(rand))
     return zero_vector(M.manifold, M.point)
 end
 
+@doc raw"""
+    base_point(TpM::TangentSpace)
+
+Return the base point of the [`TangentSpace`](@ref).
+"""
+base_point(TpM::TangentSpace) = TpM.point
+
 # forward both point checks to tangent vector checks
 function check_point(TpM::TangentSpace, p; kwargs...)
     return check_vector(TpM.manifold, TpM.point, p; kwargs...)
diff --git a/test/power.jl b/test/power.jl
index 513b113f..390efa72 100644
--- a/test/power.jl
+++ b/test/power.jl
@@ -2,11 +2,11 @@ using Test
 using ManifoldsBase
 using ManifoldsBase:
     AbstractNumbers,
-    DefaultManifold,
     ℝ,
     ℂ,
     NestedReplacingPowerRepresentation,
-    VectorSpaceType
+    VectorSpaceType,
+    DefaultManifold
 using StaticArrays
 using LinearAlgebra
 using Random
@@ -411,6 +411,16 @@ end
         @test zero_vector(N, p) == 0 .* p
     end
 
+    @testset "TangentSpace" begin
+        M = ManifoldsBase.DefaultManifold(3)
+        N = PowerManifold(M, NestedPowerRepresentation(), 2)
+        p = [[1.0, 2.0, 3.0], [4.0, 5.0, 6.0]]
+        TpN = TangentSpace(N, p)
+        Tp1M = TpN[1]
+        @test base_point(Tp1M) === p[N, 1]
+        @test base_manifold(Tp1M) === M
+    end
+
     @testset "fill" begin
         M = ManifoldsBase.DefaultManifold(3)
         N = PowerManifold(M, NestedPowerRepresentation(), 2)
diff --git a/test/product_manifold.jl b/test/product_manifold.jl
index fb516c8d..67c84266 100644
--- a/test/product_manifold.jl
+++ b/test/product_manifold.jl
@@ -1,6 +1,6 @@
 using Test
 using ManifoldsBase
-using ManifoldsBase: submanifold_component, submanifold_components
+using ManifoldsBase: DefaultManifold, submanifold_component, submanifold_components
 using ManifoldsBase:
     AbstractNumbers, ℝ, ℂ, NestedReplacingPowerRepresentation, ProductBasisData
 using LinearAlgebra
@@ -606,4 +606,10 @@ using ManifoldsBaseTestUtils
         @test ts1 × ts1 == ProductVectorTransport(tr1, tr2, tr1, tr2)
     end
 
+    @testset "TangentSpace" begin
+        TpM = TangentSpace(M, p1)
+        Tp1M1 = TpM[1]
+        @test base_point(Tp1M1) === p1[M, 1]
+        @test base_manifold(Tp1M1) === M[1]
+    end
 end