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Given a smooth submanifold of $R^n$, I was wondering if there is a reasonably simple criterion for deciding whether its normal bundle admits a flat connection. I am not ruling out monodromy in the statement of the question (thus for example, a Moebius strip immersed in $R^3$ fits the bill.)

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If I understand correctly, you are asking a purely topological question, right? You're not asking when the standard Euclidean connection on $R^n$, when restricted to the normal bundle, is flat, right? – Deane Yang Aug 8 '10 at 17:54
Yes, exactly. This is a purely topological question. – Hari Aug 10 '10 at 21:46
There is some relevant discussion here:… – Mark Grant Jan 7 '11 at 10:52

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