Book contents
- Frontmatter
- Contents
- Preface to the second edition
- Introduction
- PART A Prelude and themes: Synthetic methods and results
- PART B Development: Differential geometry
- 5 Curves in the plane
- 6 Curves in space
- 7 Surfaces
- 8 Curvature for surfaces
- 9 Metric equivalence of surfaces
- 10 Geodesics
- 11 The Gauss–Bonnet Theorem
- 12 Constant-curvature surfaces
- PART C Recapitulation and coda
- Riemann's Habilitationsvortrag: On the hypotheses which lie at the foundations of geometry
- Solutions to selected exercises
- Bibliography
- Symbol index
- Name index
- Subject index
7 - Surfaces
Published online by Cambridge University Press: 05 November 2012
- Frontmatter
- Contents
- Preface to the second edition
- Introduction
- PART A Prelude and themes: Synthetic methods and results
- PART B Development: Differential geometry
- 5 Curves in the plane
- 6 Curves in space
- 7 Surfaces
- 8 Curvature for surfaces
- 9 Metric equivalence of surfaces
- 10 Geodesics
- 11 The Gauss–Bonnet Theorem
- 12 Constant-curvature surfaces
- PART C Recapitulation and coda
- Riemann's Habilitationsvortrag: On the hypotheses which lie at the foundations of geometry
- Solutions to selected exercises
- Bibliography
- Symbol index
- Name index
- Subject index
Summary
And because of the nature of surfaces any coordinate function ought to be of two variables.
EULER, OPERA POSTHUMA (VOL. 1, P. 494)Although geometers have given much attention to general investigations of curved surfaces and their results cover a significant portion of the domain of higher geometry, this subject is still so far from being exhausted, that it can well be said that, up to this time, but a small portion of an exceedingly fruitful field has been cultivated.
GAUSS, Abstract to Disquisitiones (1827)Definition 5 of Book I of Euclid's The Elements tells us that “a surface is that which has length and breadth only.” Analytic geometry turns this definition into functions on a surface that behave like length and breadth, namely, a pair of independent coordinates that determine uniquely each point on the surface. For example, spherical coordinates, longitude and colatitude (chapter 1), apply to the surface of a sphere, and they permit new arguments via the calculus, revealing the geometry of a sphere.
Our goal in developing the classical topics of differential geometry is to discover the surfaces on which non-Euclidean geometry holds. On the way to this goal the major themes of differential geometry emerge, which include the notions of intrinsic properties, curvature, geodesics, and abstract surfaces. In this chapter we develop the analogues of notions for curves such as parameters and their transformations, tangent directions, and lengths.
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- Information
- Geometry from a Differentiable Viewpoint , pp. 116 - 155Publisher: Cambridge University PressPrint publication year: 2012