Let $M$ be a connected and simply connected compact Riemannian manifold (without boundaries). Fix a point $p\in M$.
The diameter set $D\_p$ of $p$ is the set of points that maximize the distance from $p$, i.e., $D\_p=\lbrace q : d(p,q)=\max\_r d(p,r)\rbrace$.
The cut locus $C\_p$ is the set of points at which geodesics (starting at $p$) stop to be length minimizing.


It is clear that $D\_p \subset C\_p$. What are conditions under which equality holds? Does it always hold or is it, for instance, true for symmetric spaces?



Cut locus for simply connected manifolds
The equality $D\_p = C\_p$ holds for compact symmetric spaces (CROSSes) of rank $1$, but not in general for higher rank symmetric spaces.


A rank $1$ CROSS is isometric to a round sphere or projective space with Fubini-Study metric. For a sphere, we of course have $D\_p = C\_p = \{-p\}$. On $\mathbb{K}P^n$ with $\mathbb{K}\in\{\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O}\}$, we have $D\_p = C\_p \cong \mathbb{K}P^{n-1}$.


For higher rank, consider $Sp(n)$ (the $n\times n$ quaternionic unitary matrices) with bi-invariant metric and set $p = I$, the identity matrix.


We claim that $D\_p = \{-I\}$. Indeed, given $q\in D\_p$, we may select a minimizing geodesic $\gamma$ from $p$ to $q$. This geodesic is a $1$-parameter subgroup, so can be conjugated to lie within the standard maximal torus $T^n\subseteq Sp(n)$. On the torus (which inherits its standard flat metric) clearly $-I$ is the farthest point from $I$. So, $q$ is conjugate to $-I$, which is central.


On the other hand $C\_p$ contains infinitely many points. To see this, recall that $Sp(n)\setminus \{p\}$ deformation retracts onto $C\_p$. This implies that in cohomology we have $H^3(C\_p)\neq 0$ for $n\geq 2$


