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
10 - Geodesics
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
It is as if he (al-Karābīsī) intended the notion which Archimedes expressed saying that it is the shortest distance which connects two points.
AL-NAYRĪzĪ (MS QOM 6526, 9TH-10TH CENTURY)But if a particle is not forced to move upon a determinate curve, the curve which it describes possesses a singular property, which had been discovered by metaphysical considerations; but which is in fact nothing more that a remarkable result of the preceding differential equations. It consists in this, that the integral ∫ υ ds, comprised between the two extreme points of the described curve, is less than on every other curve.
P. S. LAPLACE, MÉCANIQUE CÉLESTE (1799)To generalize other familiar geometric notions to surfaces we need a notion of “line.” Such a “line” should enjoy at least one of the elementary properties of straight lines in the plane, where lines are:
the curves of shortest length joining two points (Archimedes);
the curves of plane curvature identically zero (Huygens, Leibniz, Newton); and
the curves whose unit tangent and its derivative are linearly dependent.
In order to have geometric significance, a defining notion for a “line” on a surface must be intrinsic, that is, independent of the choice of coordinates. We first adapt the condition of zero plane curvature. Let α: (− ∈, ∈) → S be a curve on S, parametrized by arc length. The unit tangent vector is denoted by T(s) = α′(s). Consider T′(s) = α″(s); since α(s) is a unit speed curve, T(s) is perpendicular to T(s).
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- Geometry from a Differentiable Viewpoint , pp. 185 - 200Publisher: Cambridge University PressPrint publication year: 2012