## Geometry from a Differentiable ViewpointDifferential geometry has developed in many directions since its beginnings with Euler and Gauss. This often poses a problem for undergraduates: which direction should be followed? What do these ideas have to do with geometry? This book is designed to make differential geometry an approachable subject for advanced undergraduates. The text serves as both an introduction to the classical differential geometry of curves and surfaces and as a history of the non-Euclidean plane. The book begins with the theorems of non-Euclidean geometry, then introduces the methods of differential geometry and develops them towards the goal of constructing models of the hyperbolic plane. Interesting diversions are offered, such as Huygens' pendulum clock and mathematical cartography; however, the focus of the book is on the models of non-Euclidean geometry and the modern ideas of abstract surfaces and manifolds. Although the main use of this text is as an advanced undergraduate course book, the historical aspect of the text should appeal to most mathematicians. |

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### Contents

Spherical geometry | 3 |

NonEuclidean geometry I | 34 |

Curves | 63 |

Curves in space | 80 |

Surfaces | 95 |

8 Map projections | 116 |

Metric equivalence of surfaces | 145 |

Geodesics | 157 |

Constantcurvature surfaces | 186 |

Abstract surfaces | 201 |

Modeling the nonEuclidean plane | 217 |

Where from here? | 242 |

On the hypotheses which lie at the foundations | 269 |

Notes on selected exercises | 279 |

301 | |

The GaussBonnet theorem | 171 |

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abstract surface angle defect angle sum arc length asymptotic axiom Beltrami Chapter Christoffel symbols component functions compute congruent consider construct coordinate chart coordinate curves coordinate patch cos2 cosh curve a(s defined Definition denote derivative determined diffeomorphism differential equations differential geometry direction equiareal Euclid Euclidean example expression follows formula fundamental form Gauss Gaussian curvature geodesic curvature geodesically complete given line horocycle implies inner product interior angles intersection inverse isometry Lemma line element line segment linear fractional transformation manifold map projection matrix metric relations neighborhood non-Euclidean geometry parallel parametrization polar coordinates Postulate PROOF properties Proposition prove regular surface Riemann Riemann curvature tensor Riemannian metric right angles right triangle Saccheri satisfies sides space sphere straight line subset Suppose surface in R3 tangent plane tangent vector tensor Tp(S unit-speed curve vector field vertex