Straight lines on curved surfaces — geodesics reveal the geometry
A geodesic is the locally-shortest path on a curved surface — the analog of a straight line. On a sphere (K>0), geodesics are great circles that reconverge — positive curvature focuses paths. On a saddle (K<0), geodesics diverge exponentially — negative curvature defocuses. Gauss-Bonnet theorem: ∬K dA + ∮κ_g ds = 2πχ links total curvature to topology.