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A compact surface is non-orientable if and only if there is a Moebius strip on it. Is there a similar result in higher dimension?

More specifically, at least in the smooth setting, we can define a top dimensional Moebius strip as a closed loop such that the (differential of the) Poincare return map along the loop has negative determinant. Does the existence of such a closed loop is equivalent to the nonorientability of the manifold?

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Yes. If all loops give rotations as holonomy, then the holonomy group lies in the rotation group, so the manifold is orientable.

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