Tangent spaces at different points in special relativity

Question

In the book Theoretical Physics by J. C. Dutailly the author says

However because the manifolds are actually affine spaces, in SR and
Galilean Geometry the tangent spaces at different points share the same
structure (which is the underlying tangent vector space), and only in
these cases they can be assimilated to $mathbb R^4$: This is the origin of much
confusion on the subject, and the motivation to start in the GR
context where the concepts are clearly diferentiated.

My question is what does the author mean that tangent spaces at different points share the same structure?

oliver · Answer

Roughly speaking in 3D language: he means that the tangent plane of a plane is the plane itself (or rather parallel to it), and so is usually not identified as a separate entity. On the other hand, a tangent plane to a sphere is usually immediately recognized as having an important meaning because it depends on the point where the tangent plane is determined.

Answered by oliver on July 25, 2021

Tangent spaces at different points in special relativity

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