If $(X=S^n(1),d)$ is a Riemannian manifold of constant curvature, then consider two points $p,\ q\in X$ with $d(p,q)=\epsilon < \frac{\pi}{2}$.
Then what is $S:=\partial B(p,\frac{\pi}{2})\cap \partial B(q,\frac{\pi}{2})$ i.e. intersection of geodesic spheres ?
If $n=2$, then it is a union of two points. But what happen for $n\geq 3$ ?
Since ${\rm dim}\ \partial B(p,\frac{\pi}{2}) =n-1 $, then a connected component of $S$ has dimension $n-2$.
Thank you in advance.
[Add]
The connected component $C$ has dimension $n-2$. For $x_i\in C$ with small $d(x_1,x_2)$, then uniqueness of shortest geodesic and totally geodesicness of $\partial B(p,\frac{\pi}{2}) ,\ \partial B(q,\frac{\pi}{2}) $ imply that $C$ is $S^{n-2}$.
$\newcommand{\Reals}{\mathbf{R}}\newcommand{\dd}{\partial}$With distance measured in $S^{n} \subset \Reals^{n+1}$, the boundary of the $\frac{\pi}{2}$-ball centered at $p$ is $$ \dd B(p, \tfrac{\pi}{2}) = S^{n} \cap p^{\perp}, $$ the intersection of the sphere with the hyperplane $p^{\perp} \subset \Reals^{n+1}$ orthogonal to the unit vector $p$. Consequently, if $p \neq \pm q$ (i.e., $p$ and $q$ are distinct and not antipodal) then $$ \dd B(p, \tfrac{\pi}{2}) \cap \dd B(q, \tfrac{\pi}{2}) $$ is the great $(n-2)$-sphere $$ S^{n} \cap p^{\perp} \cap q^{\perp} $$ lying in the $(n - 1)$-dimensional subspace of $\Reals^{n+1}$ orthogonal to both $p$ and $q$.