If a space has curvature, then the curvature can be seen intrinsically by finding sums of angles in triangles made of geodesics. Under general relativity, space-time is curved on local scales. On global scales, experiments have determined that space-time is almost totally flat. If one were to, for each of three points in a gravitational gradient (like near a black hole), find the angle between the sight lines to the other two points and then add up all of the angles, would it add up to 180 (degrees) or perhaps it would show intrinsic curvature?
Is the way of thinking about this good, because it is quite general and conceptual. I don't actually know the equations for general relativity, but this is what I thought of when I was thinking about intrinsic geometric properties.