## Geometry from a Differentiable ViewpointThis book offers a new treatment of the topic, one which is designed to make differential geometry an approachable subject for advanced undergraduates. Professor McCleary considers the historical development of non-Euclidean geometry, placing differential geometry in the context of geometry students will be familiar with from high school. The text serves as both an introduction to the classical differential geometry of curves and surfaces and as a history of a particular surface, the non-Euclidean or hyperbolic plane. The main theorems of non-Euclidean geometry are presented along with their historical development. The author then introduces the methods of differential geometry and develops them toward the goal of constructing models of the hyperbolic plane. While interesting diversions are offered, such as Huygen's pendulum clock and mathematical cartography, the book thoroughly treats the models of non-Euclidean geometry and the modern ideas of abstract surfaces and manifolds. |

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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 |

### Common terms and phrases

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