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    Orientable manifold

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    I need to prove that the tangent bundle of a differentiable manifold is orientable even if the manifold is not. From class, all I know is that a manifold is orientable if it has an atlas {(Ua,xa)} such that when xa(Ua) intersected with xb(Ub) is non-empty then d(xb^(-1) composed with xa) has a positive determinant.

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    https://brainmass.com/math/differential-geometry/differential-geometry-orientable-tangent-bundles-204390

    Solution Summary

    This provides a proof that the tangent bundle of a differentiable manifold is orientable even if the manifold is not.

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