We realize that the extra terms involving the affine connections in the covariant derivative operator can be interpreted as excess radius due to the curvilinear nature of our space-time coordinates. We also note from the equation of geodesic motion that any frame with nonzero affine connections observes geodesic acceleration of particles in free fall. So we can make the generalization for Gaussian curvature to say that a frame of space-time has Gaussian curvature whenever it contains regions where geodesic acceleration is experienced. Now it is important to note that The affine connections can always be transformed away locally by transforming to any local free fall frame. The affine connections can be transformed away and the metric tensor reduced to that of special relativity.
Next lets consider Riemannian space-time curvature .
Lets say the two dimensional creature on the sphere carries an arrow pointing parallel to the surface in the direction of its motion as it moves from the north pole to the equator. Once it reaches the equator it moves along the equator one quarter the way around without twisting the arrow so that it stayed pointing away from the pole. The creature then returns to the pole without twisting the arrow. Upon the creatures return the creature realizes the arrow is now pointing 90 degrees away from where it initially pointed. From this the creature again infers that it is living in a universe curved within a higher dimensional space.