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Do we believe the universe has any extrinsic curvature at all? As far as I'm aware extrinsic curvature is only used in geometry/math to model the intrinsic curvature, correct?

From the answers to this question: How can we model intrinsic curvature?

Why do we still use Euclidean geometry instead of Riemannian geometry? And in what cases is one preferable over the other?

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Would there be any measurement that could reveal extrinsic curvature? I think the answer is no, which would mean that this is not a question that physics can answer (i.e. it may not make sense to even pose the question). –  nibot Apr 15 '12 at 15:03
    
On the other hand, the topology of the universe might be observable. For example, if you were able to return to your starting point by traveling very far in a single direction, we might say that the universe has "the topology of a torus". –  nibot Apr 16 '12 at 8:48
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