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