From 6f0d3e39d8239e9ac4bc571be1b61836dc4bd52a Mon Sep 17 00:00:00 2001
From: Ronny Bergmann <git@ronnybergmann.net>
Date: Fri, 24 May 2024 22:08:51 +0200
Subject: [PATCH] Fix a few typos. (#723)

---
 NEWS.md                              |  7 +++++++
 src/manifolds/SPDFixedDeterminant.jl | 13 +++++++------
 2 files changed, 14 insertions(+), 6 deletions(-)

diff --git a/NEWS.md b/NEWS.md
index 4a1716fa55..a1a19cccf6 100644
--- a/NEWS.md
+++ b/NEWS.md
@@ -5,6 +5,13 @@ 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).
 
+## [0.9.19] – unreleased
+
+### Changed
+
+* fixed a few typos in the doc string of the SPD fixed determinant description.
+
+
 ## [0.9.18] – 2024-05-07
 
 ### Added
diff --git a/src/manifolds/SPDFixedDeterminant.jl b/src/manifolds/SPDFixedDeterminant.jl
index 4b4f467e6e..0ba812a0ad 100644
--- a/src/manifolds/SPDFixedDeterminant.jl
+++ b/src/manifolds/SPDFixedDeterminant.jl
@@ -11,7 +11,9 @@ p ∈ ℝ^{n×n} \ \big|\ a^\mathrm{T}pa > 0 \text{ for all } a ∈ ℝ^{n}\back
 \bigr\}.
 ````
 
-This manifold is modelled as a submanifold of [`SymmetricPositiveDefinite`](@ref)`(n)`.
+This manifold is modelled as a submanifold of [`SymmetricPositiveDefinite`](@ref)`(n)`,
+see [`IsEmbeddedSubmanifold`](@ref `ManifoldsBase.IsEmbeddedSubmanifold`) for the implications,
+but for example retractions and inverse retractions are all available
 
 These matrices are sometimes also called [isochoric](https://en.wiktionary.org/wiki/isochoric), which refers to the interpretation of
 the matrix representing an ellipsoid. All ellipsoids that represent points on this manifold have the same volume.
@@ -21,18 +23,18 @@ and consists of all symmetric matrices with zero trace
 ```math
     T_p\mathcal P_d(n) =
     \bigl\{
-        X \in \mathbb R^{n×n} \big|\ X=X^\mathrm{T} \text{ and } \operatorname{tr}(p) = 0
+        X \in \mathbb R^{n×n} \big|\ X=X^\mathrm{T} \text{ and } \operatorname{tr}(X) = 0
     \bigr\},
 ```
-since for a constant determinant we require that `0 = D\det(p)[Z] = \det(p)\operatorname{tr}(p^{-1}Z)` for all tangent vectors ``Z``.
-Additionally we store the tangent vectors as `X=p^{-1}Z`, i.e. symmetric matrices.
+since for a constant determinant we require that ``0 = D\det(p)[Z] = \det(p)\operatorname{tr}(p^{-1}Z)`` for all tangent vectors ``Z``.
+Additionally we store the tangent vectors as ``X=p^{-1}Z``, i.e. symmetric matrices.
 
 # Constructor
 
     SPDFixedDeterminant(n::Int, d::Real=1.0; parameter::Symbol=:type)
 
 Generate the manifold $\mathcal P_d(n) \subset \mathcal P(n)$ of determinant ``d``,
-which defaults to 1.
+which defaults to `1.0`.
 
 `parameter`: whether a type parameter should be used to store `n`. By default size
 is stored in type. Value can either be `:field` or `:type`.
@@ -154,7 +156,6 @@ end
 Project the symmetric matrix `X` onto the tangent space at `p` of the
 (sub-)manifold of s.p.d. matrices of determinant `M.d` (in place of `Y`),
 by setting its diagonal (and hence its trace) to zero.
-
 """
 project(M::SPDFixedDeterminant, p, X)