Geometry from a Differentiable Viewpoint
This 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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The theory of parallels
NonEuclidean geometry I
Curves in space
gbis Map projections
Curvature for surfaces
Metric equivalence of surfaces
The GaussBonnet theorem
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abstract apply associated assumption basis called Chapter choice circle complete compute condition congruent consider constant construct containing continuous coordinate chart corresponding curvature curve defined Definition denote derivative determined differential differential equations direction distance equal Euclid Euclidean example existence expression extended fact field fixed follows formula function Gauss Gaussian curvature geodesic geometry given gives implies interior intersection introduce isometry Lemma length lies line element line segment linear manifold mapping measure meets metric non-Euclidean normal obtain origin parallel parametrization patch perpendicular plane positive Postulate projection PROOF properties Proposition prove quantities radius regular relations respect result right angles satisfies sides smooth space sphere Suppose surface tangent vector tensor theorem Tp(S transformation triangle unique unit unit-speed write
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