| 1. | Smooth structures on an orientable manifold are usually counted modulo orientation-preserving smooth homeomorphisms.
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| 2. | On non-orientable manifolds, one may instead define the weaker notion of a density.
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| 3. | This is an orientable manifold with boundary, upon which " surgery " will be performed.
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| 4. | The singular homology and cohomology groups of a closed, orientable manifold are related by Poincar?duality.
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| 5. | A manifold is orientable if it has a consistent choice of connected orientable manifold has exactly two different possible orientations.
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| 6. | A true non-orientable manifold has " closed paths " that take travelers from R to / .
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| 7. | The most familiar example is orientability : some manifolds are orientable, some are not, and orientable manifolds admit 2 orientations.
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| 8. | See also orientable manifold ( the article may be a bit advanced, but it formally pinpoints some of the above observations ).
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| 9. | An orientable manifold has infinitely many volume forms, since multiplying a volume form by a non-vanishing function yields another volume form.
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| 10. | Both bundles are 2-manifolds, but the annulus is an orientable manifold while the M�bius band is a non-orientable manifold.
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