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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
up vote 3 down vote accepted

Ambient spaces can always be defined in the sense that, by the Nash embedding theorem, every Riemannian manifold isometrically embeds into some Euclidean space. However, there is no guarantee that something you define in terms of a choice of embedding is independent of the choice of embedding, and mathematicians have found that it is cleaner to separate out what can be talked about without a choice of embedding vs. what can be talked about without such a choice.

In practice, one also comes across Riemannian manifolds which don't admit a natural choice of embedding, so one is forced to choose one in some unnatural way if one does everything in terms of embeddings. Examples include quotients by a group action.

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Not to mention the fact that Nash's embedding theorem is rather a difficult and intricate piece of mathematics that came well after the definition of Riemannian manifold. – treble Jan 1 '13 at 4:19

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