diff --git a/NEWS.md b/NEWS.md
index 931ff41dda..b7bf9f8f4c 100644
--- a/NEWS.md
+++ b/NEWS.md
@@ -5,6 +5,12 @@ 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).
 
+## [0.9.7] - 2023-11-14
+
+### Fixed
+
+- Fixed `is_flat` for `CholeskySpace` and `SymmetricPositiveDefinite` with `LogCholeskyMetric` [https://github.com/JuliaManifolds/Manifolds.jl/issues/684](https://github.com/JuliaManifolds/Manifolds.jl/issues/684).
+
 ## [0.9.6] - 2023-11-09
 
 ### Fixed
diff --git a/Project.toml b/Project.toml
index 932393dddd..50ae539bec 100644
--- a/Project.toml
+++ b/Project.toml
@@ -1,7 +1,7 @@
 name = "Manifolds"
 uuid = "1cead3c2-87b3-11e9-0ccd-23c62b72b94e"
 authors = ["Seth Axen <seth.axen@gmail.com>", "Mateusz Baran <mateuszbaran89@gmail.com>", "Ronny Bergmann <manopt@ronnybergmann.net>", "Antoine Levitt <antoine.levitt@gmail.com>"]
-version = "0.9.6"
+version = "0.9.7"
 
 [deps]
 Distributions = "31c24e10-a181-5473-b8eb-7969acd0382f"
diff --git a/src/manifolds/CholeskySpace.jl b/src/manifolds/CholeskySpace.jl
index ff3b9af276..5127235a07 100644
--- a/src/manifolds/CholeskySpace.jl
+++ b/src/manifolds/CholeskySpace.jl
@@ -132,9 +132,9 @@ end
 """
     is_flat(::CholeskySpace)
 
-Return false. [`CholeskySpace`](@ref) is not a flat manifold.
+Return true. [`CholeskySpace`](@ref) is a flat manifold. See Proposition 8 of [Lin:2019](@cite).
 """
-is_flat(M::CholeskySpace) = false
+is_flat(M::CholeskySpace) = true
 
 @doc raw"""
     log(M::CholeskySpace, X, p, q)
diff --git a/src/manifolds/SymmetricPositiveDefiniteLogCholesky.jl b/src/manifolds/SymmetricPositiveDefiniteLogCholesky.jl
index 25c7d22d42..47c43ea723 100644
--- a/src/manifolds/SymmetricPositiveDefiniteLogCholesky.jl
+++ b/src/manifolds/SymmetricPositiveDefiniteLogCholesky.jl
@@ -99,10 +99,10 @@ end
 """
     is_flat(::MetricManifold{ℝ,<:SymmetricPositiveDefinite,LogCholeskyMetric})
 
-Return false. [`SymmetricPositiveDefinite`](@ref) with [`LogCholeskyMetric`](@ref)
-is not a flat manifold.
+Return true. [`SymmetricPositiveDefinite`](@ref) with [`LogCholeskyMetric`](@ref)
+is a flat manifold. See Proposition 8 of [Lin:2019](@cite).
 """
-is_flat(M::MetricManifold{ℝ,<:SymmetricPositiveDefinite,LogCholeskyMetric}) = false
+is_flat(M::MetricManifold{ℝ,<:SymmetricPositiveDefinite,LogCholeskyMetric}) = true
 
 @doc raw"""
     log(M::MetricManifold{ℝ,<:SymmetricPositiveDefinite,LogCholeskyMetric}, p, q)
diff --git a/src/manifolds/SymmetricPositiveSemidefiniteFixedRank.jl b/src/manifolds/SymmetricPositiveSemidefiniteFixedRank.jl
index 9be8593caf..0f096829b7 100644
--- a/src/manifolds/SymmetricPositiveSemidefiniteFixedRank.jl
+++ b/src/manifolds/SymmetricPositiveSemidefiniteFixedRank.jl
@@ -156,7 +156,7 @@ end
 """
     is_flat(::SymmetricPositiveSemidefiniteFixedRank)
 
-Return false. [`SymmetricPositiveSemidefiniteFixedRank`](@ref) is not a flat manifold.
+Return false. [`SymmetricPositiveSemidefiniteFixedRank`](@ref) is not a flat manifold. See Theorem A.18 in [MassartAbsil:2020](@cite).
 """
 is_flat(M::SymmetricPositiveSemidefiniteFixedRank) = false
 
diff --git a/test/manifolds/cholesky_space.jl b/test/manifolds/cholesky_space.jl
index 2cbd2e11db..3fe165f654 100644
--- a/test/manifolds/cholesky_space.jl
+++ b/test/manifolds/cholesky_space.jl
@@ -4,7 +4,7 @@ include("../utils.jl")
     M = Manifolds.CholeskySpace(3)
     @test repr(M) == "CholeskySpace(3)"
 
-    @test !is_flat(M)
+    @test is_flat(M)
 
     types = [Matrix{Float64}]
     TEST_FLOAT32 && push!(types, Matrix{Float32})
diff --git a/test/manifolds/symmetric_positive_definite.jl b/test/manifolds/symmetric_positive_definite.jl
index 254f187f69..e8d0e6c091 100644
--- a/test/manifolds/symmetric_positive_definite.jl
+++ b/test/manifolds/symmetric_positive_definite.jl
@@ -34,7 +34,11 @@ include("../utils.jl")
         end
         @testset "$(typeof(M))" begin
             @test representation_size(M) == (3, 3)
-            @test !is_flat(M)
+            if M === M3
+                @test is_flat(M)
+            else
+                @test !is_flat(M)
+            end
             for T in types
                 exp_log_atol_multiplier = 8.0
                 if T <: MMatrix{3,3,Float64}