## A Comprehensive Introduction to Differential Geometry, Volume 3 |

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### Contents

The Fundamental Equations for Hypersurfaces | 1 |

Covariant differentiation in a submanifold of a Riemannian manifold | 8 |

The moving frame description | 23 |

Copyright | |

22 other sections not shown

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### Common terms and phrases

actually angle assume asymptotic curve basis called Chapter choice choose circle clearly closed compact complete component condition connected Consequently consider constant contains coordinate Corollary corresponding curvature curve defined definition depends derivative determined developable direction equation everywhere fact follows formulas function fundamental geodesic given gives hence immersion integral intersect invariant inversion k₁ k₂ Lemma LIBRARIES lies lines lines of curvature manifold matrix means metric MICHIGAN moving frame neighborhood normal obtain oriented orthonormal parallel parameterized perpendicular plane positively principal Problem Proof Proposition prove respect result Riemannian ruled satisfies shows side similarity simply space sphere straight lines Suppose surface tangent tangent vector Theorem umbilics unit vector UNIVERSITY vector field Volume zero