orientable manifold造句

• Smooth structures on an orientable manifold are usually counted modulo orientation-preserving smooth homeomorphisms.
• On non-orientable manifolds, one may instead define the weaker notion of a density.
• This is an orientable manifold with boundary, upon which " surgery " will be performed.
• The singular homology and cohomology groups of a closed, orientable manifold are related by Poincar?duality.
• A manifold is orientable if it has a consistent choice of connected orientable manifold has exactly two different possible orientations.
• A true non-orientable manifold has " closed paths " that take travelers from R to / .
• The most familiar example is orientability : some manifolds are orientable, some are not, and orientable manifolds admit 2 orientations.
• See also orientable manifold ( the article may be a bit advanced, but it formally pinpoints some of the above observations ).
• An orientable manifold has infinitely many volume forms, since multiplying a volume form by a non-vanishing function yields another volume form.
• Both bundles are 2-manifolds, but the annulus is an orientable manifold while the M鯾ius band is a non-orientable manifold.
• Both bundles are 2-manifolds, but the annulus is an orientable manifold while the M鯾ius band is a non-orientable manifold.
• The Desargues graph can be embedded as a self-regular map in the non-orientable manifold of genus 6, with decagonal faces.
• 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.
• 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.
• 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 ).
• 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 .,
• 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.
• 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.
• 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 .)
• 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.