| 11. | The connected manifold is the same as that of the manifold.
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| 12. | Such an object can exist in certain Euclidean 3-manifolds.
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| 13. | The daisy valve is bolted to the manifold through its centre.
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| 14. | The study of Riemannian manifolds constitutes the subject called Riemannian geometry.
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| 15. | Calabi & ndash; Yau manifolds are important in superstring theory.
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| 16. | An analogous criterion can be formulated for maps between differentiable manifolds.
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| 17. | Spacetime can be modeled as a 4-dimensional Lorentzian manifold.
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| 18. | Calabi-Yau manifolds that give rise to the same physics.
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| 19. | This larger manifold is called the phase space of the system.
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| 20. | This is done by defining coordinate functions on this new manifold as
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