I am reading Kuhnel's Differential Geometry of Curves,Surfaces,Manifolds (2ed). On p.209, discussing tangent space of riemannian manifold, it says: ``since there is no ambient space, this notion has to be intrinsically defined''. Does this mean there is never an ambient space, or just that this branch of geometry endevours to not make use of the ambient space even when it exists?
If the former, is it easy to give an example of a situation where no ambient space can be defined?