Both bundles are 2-manifolds, but the annulus is an orientable manifold while the M�bius band is a non-orientable manifold.
12.
The Desargues graph can be embedded as a self-regular map in the non-orientable manifold of genus 6, with decagonal faces.
13.
The cohomology with local coefficients in the module corresponding to the orientation covering can be used to formulate Poincar?duality for non-orientable manifolds : see Twisted Poincar?duality.
14.
The 5-regular Clebsch graph can be embedded as a regular map in the orientable manifold of genus 5, forming pentagonal faces; and in the non-orientable surface of genus 6, forming tetragonal faces.
15.
Every closed 3-manifold has a prime decomposition : this means it is the connected sum of prime 3-manifolds ( this decomposition is essentially unique except for a small problem in the case of non-orientable manifolds ).
16.
Notice the striking similarity between this statement and the generalized version of Stokes'theorem, which says that the integral of any orientable manifold ? is equal to the integral of its exterior derivative d? over the whole of ?, i . e .,
17.
Restricting to changes of coordinates with positive Jacobian determinant is possible on orientable manifolds, because there is a consistent global way to eliminate the minus signs; but otherwise the line bundle of densities and the line bundle of " n "-forms are distinct.
18.
It is also possible to work directly with non-orientable manifolds, but this gives some extra complications : it may be necessary to cut along projective planes and Klein bottles as well as spheres and tori, and manifolds with a projective plane boundary component usually have no geometric structure.
19.
On non-orientable manifolds this identification cannot be made, since the density bundle is the tensor product of the orientation bundle of " M " and the " n "-th exterior product bundle of " T * M " ( see pseudotensor .)
20.
This also works for non-orientable manifolds, which have a \ mathbf { Z } / 2 \ mathbf { Z }-orientation, in which case one obtains \ mathbf { Z } / 2 \ mathbf { Z }-valued characteristic numbers, such as the Stiefel-Whitney numbers.
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