diff --git a/NEWS.md b/NEWS.md
index cf148c70..ddf7922b 100644
--- a/NEWS.md
+++ b/NEWS.md
@@ -5,6 +5,11 @@ 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] unreleased
+
+* 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
 
 ### Added
diff --git a/Project.toml b/Project.toml
index 1c1fe3ca..a2f5afa3 100644
--- a/Project.toml
+++ b/Project.toml
@@ -1,7 +1,7 @@
 name = "ManifoldsBase"
 uuid = "3362f125-f0bb-47a3-aa74-596ffd7ef2fb"
 authors = ["Seth Axen <seth.axen@gmail.com>", "Mateusz Baran <mateuszbaran89@gmail.com>", "Ronny Bergmann <manopt@ronnybergmann.net>", "Antoine Levitt <antoine.levitt@gmail.com>"]
-version = "0.15.9"
+version = "0.15.10"
 
 [deps]
 LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
diff --git a/src/ManifoldsBase.jl b/src/ManifoldsBase.jl
index 09054298..af7877cf 100644
--- a/src/ManifoldsBase.jl
+++ b/src/ManifoldsBase.jl
@@ -1219,6 +1219,7 @@ export ×,
     allocate,
     angle,
     base_manifold,
+    base_point,
     change_basis,
     change_basis!,
     change_metric,
diff --git a/src/PowerManifold.jl b/src/PowerManifold.jl
index 78daf96b..39c96518 100644
--- a/src/PowerManifold.jl
+++ b/src/PowerManifold.jl
@@ -778,6 +778,23 @@ 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 a [`AbstractPowerManifold`](@ref)s tangent space `TpM`.
+"""
+function Base.getindex(
+    TpM::TangentSpace{𝔽,<:AbstractPowerManifold},
+    I::Union{Integer,Colon,AbstractVector}...,
+) where {𝔽}
+    M = base_manifold(TpM).manifold
+    p = base_point(TpM)[base_manifold(TpM), I]
+    return TangentSpace(M, p)
+end
+
+
+
 @doc raw"""
     injectivity_radius(M::AbstractPowerManifold[, p])
 
diff --git a/src/ProductManifold.jl b/src/ProductManifold.jl
index f5b4b80c..4fbe3719 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)
+    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..2b20d33d 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"""
+    bae_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 aed5c441..b0e06bff 100644
--- a/test/power.jl
+++ b/test/power.jl
@@ -388,4 +388,13 @@ struct TestArrayRepresentation <: AbstractPowerRepresentation end
         p = rand(N)
         @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 = TangentSpace(M, p[N, 1])
+        @test base_point(Tp1M) === p[1]
+    end
 end
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