# in Riemannian geometry, when is there an ambient space?

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?

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The use of the word "is" in this sentence is misleading. "Has an ambient space" is not a property, it's a structure. – Qiaochu Yuan Dec 31 '12 at 6:24
I interpret the quoted sentence as "since there has not been introduced an ambient space, ..." – user53153 Dec 31 '12 at 6:41