Advertisement

A Brief Guide to the Local Theory of Surfaces in R3

  • Marcel Berger
  • Bernard Gostiaux
Chapter
  • 2.7k Downloads
Part of the Graduate Texts in Mathematics book series (GTM, volume 115)

Abstract

The local study of curves in R2, in sections 4 and 5 of chapter 9, was simple: theorem 8.5.7 shows that a single invariant, the curvature (expressed as a function of the arclength), is enough to characterize such a curve. The fundamental reason for this simplicity, and one that remains true no matter what the dimension of the ambient space, is that the intrinsic geometry of curves is trivial; the metric given by the length of paths on a curve is always the same as the metric on some interval of R. And the “shape”, or “position,” of a curve in R2 is specified by a mere scalar function, the curvature.

Keywords

Minimal Surface Fundamental Form Gaussian Curvature Local Theory Parallel Transport 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer Science+Business Media New York 1988

Authors and Affiliations

  • Marcel Berger
    • 1
  • Bernard Gostiaux
    • 2
  1. 1.I.H.E.S.Bures-sur-YvetteFrance
  2. 2.Le PerreuxFrance

Personalised recommendations