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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 $S^{n-2}$.

HK Lee
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    Is distance measured along the sphere, or chordally in the ambient space? On the assumption of the former, is $0 < \epsilon < \pi$ (i.e., $p$ and $q$ are distinct and not antipodal, and the respective boundaries are the great hyperspheres orthogonal to $p$ and $q$)? – Andrew D. Hwang May 11 '17 at 13:52
  • I add more conditions – HK Lee May 11 '17 at 13:58

1 Answers1

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$\newcommand{\Reals}{\mathbf{R}}\newcommand{\dd}{\partial}$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 $q$.

Andrew D. Hwang
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