A Brief Guide to the Local Theory of Surfaces in R3

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


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.


Minimal Surface Fundamental Form Gaussian Curvature Local Theory Parallel Transport 
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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

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