| 41. | A Riemannian manifold is a differentiable manifold on which the tangent spaces are equipped with inner products in a differentiable fashion.
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| 42. | Completeness here is understood in the sense that the exponential map is defined on the whole tangent space of a point.
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| 43. | In some cases, one might like to have a direct definition of the cotangent space without reference to the tangent space.
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| 44. | The ( local ) de Rham isomorphism follows by continuing this process until a complete reduction of the tangent space is achieved:
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| 45. | The metric tensor g _ { \ alpha \ beta } \ ! gives the inner product in the tangent space directly:
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| 46. | A manifold which admits a smooth choice of orientations for its tangents spaces is said to be " orientable ".
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| 47. | For example, the blowup of a point in a plane replaces the point with the projectivized tangent space at that point.
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| 48. | Quadratic differentials on a Riemann surface X are identified with the tangent space at ( X, f ) to Teichm�ller space.
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| 49. | A frame ( or, in more precise terms, a tangent frame ) is an ordered basis of particular tangent space.
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| 50. | The tangent space at " p " is isometric as a real inner product space to E 1, 3.
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