Titlebook: Introduction to Hyperbolic Geometry; Arlan Ramsay,Robert D. Richtmyer Textbook 1995 Springer Science+Business Media New York 1995 derivati

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Differential Geometry of Surfaces,This chapter is intended to help the reader learn some of the concepts of differential geometry as they appear in terms of coordinates and also to move toward thinking of them in intrinsic terms (‘coordinate free’), which is the modern approach to differential geometry. Our primary interest is in the hyperbolic plane as a Riemannian manifold.

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https://doi.org/10.1007/978-1-4757-5585-5derivation; differential equation; differential geometry; differential geometry of surfaces; hyperbolic

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978-0-387-94339-8Springer Science+Business Media New York 1995

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Textbook 1995tandard upper division course on hyperbolic geometry. For that material, the students need to be familiar with calculus and linear algebra and willing to accept one advanced theorem from analysis without proof. The book goes well beyond the standard course in later chapters, and there is enough mate

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0172-5939 ok for a standard upper division course on hyperbolic geometry. For that material, the students need to be familiar with calculus and linear algebra and willing to accept one advanced theorem from analysis without proof. The book goes well beyond the standard course in later chapters, and there is e

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Axioms for Plane Geometry,s consistency and its uniqueness.) Our axiom system is equivalent to that of Hilbert for the hyperbolic plane, and following the laying of the foundations in this chapter we proceed rigorously with the development of its properties, its consistency and its uniqueness, in later chapters.

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Qualitative Description of the Hyperbolic Plane,ed so closely to the isometries of the plane, they are also very different from the Euclidean case. All of the distinctive properties can be traced to the hyperbolic parallel axiom, so we begin with the theory of parallelism, presented in roughly the form in which Gauss derived it.

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Consistency and Categoricalness of the Hyperbolic Axioms; The Classical Models,It follows that the hyperbolic axioms are consistent, if the axioms of ? are consistent. It is proved that the axiom system is categorical, in the that any model of the hyperbolic plane is isomorphic to any other model. Lastly, as an amusement, we describe a hyperbolic model of the Euclidean plane.