| 21. | Each point of an " n "-dimensional differentiable manifold has a tangent space.
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| 22. | As a result, tangent spaces and vectors are defined as operators acting on this space of functions.
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| 23. | This fact is utilized by employing tetrad fields describing a flat tangent space at every point of spacetime.
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| 24. | Another way of saying this is that the pullback of the metric onto the tangent space is degenerate.
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| 25. | Slerp curves not extending through a point fail to transform into lines in that point's tangent space.
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| 26. | The differential or pushforward of a map between manifolds is the induced map between tangent spaces of those maps.
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| 27. | SU ( 2 ) _ R, and hence the quaternions act upon the tangent space of extended superspace.
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| 28. | This formulation is analogous to the construction of the cotangent space to define the Zariski tangent space in algebraic geometry.
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| 29. | More strictly this defines an affine tangent space, distinct from the space of tangent vectors described by modern terminology.
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| 30. | This yields the correspondence between the tangent space defined via derivations and the tangent space defined via the cotangent space.
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