Editing Orientability (section)

===Orientability of differentiable manifolds===

The most intuitive definitions require that ''M'' be a differentiable manifold.  This means that the transition functions in the atlas of ''M'' are ''C''<sup>1</sup>-functions.  Such a function admits a [[Jacobian determinant]].  When the Jacobian determinant is positive, the transition function is said to be '''orientation preserving'''.  An '''oriented atlas''' on ''M'' is an atlas for which all transition functions are orientation preserving.  ''M'' is '''orientable''' if it admits an oriented atlas.  When {{math|''n'' &gt; 0}}, an '''orientation''' of ''M'' is a maximal oriented atlas.  (When {{math|1=''n'' = 0}}, an orientation of ''M'' is a function {{math|''M'' → {±1}<nowiki/>}}.)

Orientability and orientations can also be expressed in terms of the tangent bundle.  The tangent bundle is a [[vector bundle]], so it is a [[fiber bundle]] with [[structure group]] {{math|GL(''n'', '''R''')}}.  That is, the transition functions of the manifold induce transition functions on the tangent bundle which are fiberwise linear transformations.  If the structure group can be reduced to the group {{math|GL<sup>+</sup>(''n'', '''R''')}} of positive determinant matrices, or equivalently if there exists an atlas whose transition functions determine an orientation preserving linear transformation on each tangent space, then the manifold ''M'' is orientable.  Conversely, ''M'' is orientable if and only if the structure group of the tangent bundle can be reduced in this way.  Similar observations can be made for the frame bundle.

Another way to define orientations on a differentiable manifold is through [[volume form]]s.  A volume form is a nowhere vanishing section ''&omega;'' of {{math|⋀{{sup|''n''}} ''T''{{i sup|∗}}''M''}}, the top exterior power of the cotangent bundle of ''M''.  For example, '''R'''<sup>''n''</sup> has a standard volume form given by {{math|''dx''<sup>1</sup> ∧ ⋯ ∧ ''dx''<sup>''n''</sup>}}.  Given a volume form on ''M'', the collection of all charts {{math|''U'' → '''R'''<sup>''n''</sup>}} for which the standard volume form pulls back to a positive multiple of ''&omega;'' is an oriented atlas.  The existence of a volume form is therefore equivalent to orientability of the manifold.

Volume forms and tangent vectors can be combined to give yet another description of orientability.  If {{math|''X''<sub>1</sub>, …, ''X''<sub>''n''</sub>}} is a basis of tangent vectors at a point ''p'', then the basis is said to be '''right-handed''' if {{math|&omega;(''X''<sub>1</sub>, …, ''X''<sub>''n''</sub>) &gt; 0}}.  A transition function is orientation preserving if and only if it sends right-handed bases to right-handed bases.  The existence of a volume form implies a reduction of the structure group of the tangent bundle or the frame bundle to {{math|GL<sup>+</sup>(''n'', '''R''')}}.  As before, this implies the orientability of ''M''.  Conversely, if ''M'' is orientable, then local volume forms can be patched together to create a global volume form, orientability being necessary to ensure that the global form is nowhere vanishing.