Suppose in order to probe the curvature of a manifold in $\Bbb R^2$, you decide to shoot off geodesics from the four corners of the square.
Consider a set of geodesics, $K,$ each symmetrical about the line $y=1-x,$ embedded in the unit square, that pass through the points $(0,0)$ and $(1,1):$
$$ K=\{s_1,s_2,s_3,...\}, $$
such that, $$ s_1>s_2>s_3>... $$
Furthermore, at the boundary of the square, the distance between successive geodesics goes to zero.
Define a new set of geodesics, $G,$ each symmetrical about the line $y=x,$ embedded in the unit square, that pass through the points, $(0,0)$ and $(1,1):$
$$ G=\{t_1,t_2,t_3,...\},$$
such that,
$$ ...>t_2>t_1>s_1>s_2>...$$
At the boundary, the distance between successive geodesics goes to zero.
Define a new set $R=\{K,G\},$ that is the union of the sets $K$ and $G.$
Define a new set $B$ as the reflection of $R$ about the line $x=1/2,$ where all curves in $B$ are also geodesics.
Suppose each geodesic runs across a manifold, $M,$ in $\Bbb R^2.$
What is the curvature of the manifold?
What other information can one recover about the manifold?
This is the result of probing a particular manifold with geodesics:
