Orientability
= Orientability
{wiki}
Orientability asks whether local choices of handedness can agree everywhere on a <manifold>. On a <smooth manifold>, an <orientation of a smooth manifold> chooses an <orientation of a vector space> for each <tangent space>, continuously with position. An <ordered basis> describes the same handedness as another one when the change-of-basis <determinant> is positive. An <oriented atlas> implements this condition through positive <Jacobian determinants> of the <smooth transition maps>.
A paper strip makes the obstruction visible. Join its short ends without twisting: the resulting <untwisted band> admits a consistent choice of <unit normal>. Add a half-turn before joining: on the <Möbius band>, following the <core circle of a Möbius band> reverses that choice when you first return to the same point. A local decision about handedness has encountered a global contradiction. The <normal reversal around the Möbius core> makes this precise for an explicit model.
\Image[/orientability-cylinder-mobius-edge-gluing.png]
{title=Matching and reversed edge identifications on a cylinder and a Möbius band}
Both rectangles have coordinates $(u,v)\in[0,1]\times[-1,1]$. The <untwisted band> identifies $(0,v)$ with $(1,v)$; the <Möbius band> identifies $(0,v)$ with $(1,-v)$. Across a seam the corresponding local coordinate changes preserve or reverse $v$, giving <determinants> $+1$ and $-1$. This sign explains the handedness change; the drawing also tracks the two <boundary> circles of the <untwisted band> and the single <boundary> circle of the <Möbius band>.
For a <smooth> <topological surface> embedded in $\mathbb R^3$, choosing an <orientation of a surface> is equivalent to choosing a continuous <unit normal>. This picture is specific to surfaces in three-dimensional space: the general definition uses compatible local orientations, without requiring an <embedding>. <Orientability> is a <topological invariant>, so a <homeomorphism> cannot turn a <Möbius band> into a <untwisted band>. The <sphere> and <torus> are <orientable surfaces>; the <Klein bottle> is not. https://www.open.edu/openlearn/science-maths-technology/mathematics-statistics/surfaces/content-section-3.2[Open University, Surfaces, §3.2: orientability]{ref}
The distinction matters when a sign has geometric meaning. A signed <flux> <surface integral> needs an <orientation of a surface>, and <Stokes theorem> relates that choice to the <outward-normal-first boundary orientation>. Ordinary unsigned <surface area> does not require <orientability>. Thus a <Möbius band> still has a perfectly meaningful area even though it has no continuous global <unit normal>.
A <connected> orientable <manifold> of positive dimension has two possible orientations: reversing every local choice gives the other one. If a <manifold> has several <connected components>, each component can be oriented independently when it is orientable. For <topological surfaces>, a useful obstruction criterion is the presence of a <Möbius band>: a surface is nonorientable exactly when it contains such a band. https://pi.math.cornell.edu/~mec/Winter2009/Victor/part3.htm[Cornell Mathematical Explorers' Club, Orientable and Nonorientable Surfaces]{ref}