The surface of a sphere embedded in Euclidean space has positive curvature and (eponymous) spherical geometry.
But what if I construct an $n$-sphere (e.g., a 2-sphere), defined as the set of points equidistant from some center point, in a negatively-curved $n+1$-dimensional hyperbolic space (e.g., $H^3$)? Does such a surface still always have positive curvature?
In other words, would a person walking around on a planet in a hyperbolic universe feel like they were walking around on a normal Euclidean planet, or would they feel like they were getting lost in a game of HyperRogue? Or does it depend on the size of the planet?
Best Answer
Let me describe steps for proving this for the 2-dimensional hyperbolic spheres; I will leave you the higher dimensional case as an exercise:
Prove that all 2-dimensional geodesic spheres of the same radius in $H^3$ are congruent to each other.
Working with the unit ball model of $H^3$ check that $O(3)$ is the stabilizer of the center $0$ of the unit ball in the isometry group of $H^3$.
Check that $O(3)$ acts transitively on every hyperbolic sphere $S(0,R)$ of radius $R$ centered at $0$.
Using 3, prove that every $S(0,R)$ has constant Gaussian curvature.
Use the Gauss-Bonnet formula, or Killing-Hopf theorem or other tools you know, to conclude that $S(0,R)$ has positive curvature.
If you like, I can give an explicit curvature formula using hyperbolic trigonometry.