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?

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.