How can inhabitants of a curved surface discover curvature without leaving their surface?

Inhabitants of a curved surface can determine its curvature without leaving the surface by utilizing intrinsic geometry, a concept famously formalized by Carl Friedrich Gauss.

While extrinsic curvature describes how a surface bends within a higher-dimensional space, intrinsic curvature—specifically Gaussian curvature—can be measured entirely from within the surface itself. This is established by Gauss’s "Remarkable Theorem," which proves that Gaussian curvature is an intrinsic property that remains unchanged even if the surface is bent without stretching.1

Inhabitants can detect this curvature through several methods:

• Geodesic Deviation: By measuring the behavior of geodesics—the shortest paths between points—inhabitants can identify curvature. On a flat surface, parallel lines remain equidistant. On a surface with positive curvature (like a sphere), initially parallel geodesics will eventually converge, while on a surface with negative curvature (like a hyperbolic plane), they will diverge.1
• Geometric Measurements: Inhabitants can measure the properties of shapes drawn on their surface. For example, the sum of the interior angles of a triangle will deviate from 180 degrees depending on the surface's Gaussian curvature.1

These measurements allow inhabitants to distinguish between flat, spherical, and hyperbolic geometries without ever needing an external perspective.1