# Orientable manifold

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.

© BrainMass Inc. brainmass.com October 9, 2019, 9:59 pm ad1c9bdddfhttps://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.

$2.19