In a space of constant curvature $K$, the function for $C(r)$ where $C$ is the circumference of a circle of radius $r$ satisfies:
$C''=−KC$, with initial conditions $C(0)=0$ and $C'(0)=2\pi$.
(Units check: Gaussian curvature $K$ is units $1/\rm length^2$, $C$ is units $\rm length$, and $C''$ is units $1/\rm length$.)
When $K=0$, as in flat space, this gives us the familiar $C=2\pi r$ that we all know and love. When $K>0$ this gives us $C=2\pi\frac{\sin(r\sqrt K)}{\sqrt K}$, and when $K<0$ this gives us $C=2\pi\frac{\sinh (r\sqrt{-K})}{\sqrt{-K}}$.
Is there an intuitive reason why $C''=-KC$ holds? In other words, is there an intuitive reason why $C''/C$ is constant in surfaces of constant curvature? (The initial conditions $C(0)=0$ and $C'(0)=2\pi$ are, I think, pretty intuitive.)
I think the explanation lies in Jacobi fields, but I forget the details of how those work.
We can frame the question in a different way. Given a point $p$ and any function $C$, we can "grow" a manifold around $p$ such that $C$ is the circumference of circles of radius $r$ around $p$. This gives us a whole family of manifolds, including for example the paraboloid, but the resulting manifold only looks the same at every point when $C''/C$ is constant. (You can make circles grow like $C$ from a given point, but unless $C''/C$ is constant, you'll have trouble making them grow like that from every point.) Why?