Final answer:
The analysis of geodesics and projection of their separation vectors should yield the same result whether performed intrinsically or extrinsically, because intrinsic curvature dictates geodesic behavior on a surface. However, extrinsic calculations may involve additional components from the embedding space.
Step-by-step explanation:
The analysis of geodesics on a surface and their separation vectors is a topic within the field of differential geometry, which is part of higher-level mathematics. When working with geodesics on a surface, an intrinsic view considers properties that are dependent only on the surface itself, not on how the surface is embedded in space. On the other hand, an extrinsic view would take into account how the surface is situated within the ambient space. In the intrinsic approach, the curvature and other properties are understood strictly from within the surface, as understood by an inhabitant of the surface who cannot perceive the external space.
Projecting the double derivative of the separation vector of two geodesics onto the surface is an operation that involves both intrinsic and extrinsic properties. In theory, if the analysis is correctly performed intrinsically, it should yield the same result as when the projection is considered from an extrinsic perspective, since the intrinsic curvature dictates geodesic behavior. However, the calculation from an embedded point of view may involve additional components related to the embedding space which do not affect the intrinsic geometry.