## A comprehensive introduction to differential geometry, Volume 2 |

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### Contents

MANIFOLDS 1l to 132 | 1-1 |

CURVES IN THE PLANE AND IN SPACE 1l to 16l | 1-15 |

CHAPTER UA All INAUGURAL LECTURE UA1 to UA20 | 1-20 |

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### Common terms and phrases

1-forms angle arbitrary ay ay calculation Chapter choose Christoffel symbols classical clearly components connection Consequently consider constant curvature convex coordinate system Corollary covariant derivative curvature tensor define definition denote diffeomorphism ellipsoid equivalent Euclidean motion expression extended manifold follows formula function f Gauss Gaussian curvature geodesic geometry given hence identity implies independent inner product invariant isometry Lemma line element linear locally isometric matrix metric relations moving frame neighborhood numbers obtain orthonormal osculating plane parallel translation parameterized by arclength perpendicular points c(s principal bundle Proof Proposition prove quadratic function result Riemann Riemannian manifold Riemannian metric Riemannian normal coordinate satisfy shows spanned structural equation surface tangent line tangent space tangent vector tensor of type Theorem Theorema Egregium torsion transformation unique unit vector usual Riemannian metric vector field vector space x-y plane