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Orientability
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==Orientation of vector bundles== {{main|Orientation of a vector bundle}} {{See also|Euler class}} A real [[vector bundle]], which ''a priori'' has a [[GL(n)]] [[structure group]], is called ''orientable'' when the [[structure group]] may be [[Reduction of the structure group|reduced]] to <math>GL^{+}(n)</math>, the group of [[matrix (mathematics)|matrices]] with positive [[determinant]]. For the [[tangent bundle]], this reduction is always possible if the underlying base manifold is orientable and in fact this provides a convenient way to define the orientability of a [[smooth function|smooth]] real [[manifold]]: a smooth manifold is defined to be orientable if its [[tangent bundle]] is orientable (as a vector bundle). Note that as a manifold in its own right, the tangent bundle is ''always'' orientable, even over nonorientable manifolds.
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