| 1. | See the article on orientability for more on orientations of manifolds.
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| 2. | Thus the existence of a volume form is equivalent to orientability.
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| 3. | Orientability and orientations can also be expressed in terms of the tangent bundle.
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| 4. | It is possible to drop the orientability condition and work with coefficients instead.
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| 5. | The notion of orientability can be generalised to higher-dimensional manifolds as well.
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| 6. | The existence of a volume form is therefore equivalent to orientability of the manifold.
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| 7. | In other words, Euler characteristic and orientability completely classify closed surfaces up to homeomorphism.
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| 8. | Volume forms and tangent vectors can be combined to give yet another description of orientability.
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| 9. | In Lorentzian geometry, there are two kinds of orientability : space orientability and time orientability.
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| 10. | In Lorentzian geometry, there are two kinds of orientability : space orientability and time orientability.
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