Differential invariants of surfaces

Peter J. Olver

Research output: Contribution to journalArticlepeer-review

19 Scopus citations


The algebra of differential invariants of a suitably generic surface S ⊂ R3, under either the usual Euclidean or equi-affine group actions, is shown to be generated, through invariant differentiation, by a single differential invariant. For Euclidean surfaces, the generating invariant is the mean curvature, and, as a consequence, the Gauss curvature can be expressed as an explicit rational function of the invariant derivatives, with respect to the Frenet frame, of the mean curvature. For equi-affine surfaces, the generating invariant is the third order Pick invariant. The proofs are based on the new, equivariant approach to the method of moving frames.

Original languageEnglish (US)
Pages (from-to)230-239
Number of pages10
JournalDifferential Geometry and its Application
Issue number2
StatePublished - Apr 2009


  • Differential invariant
  • Equi-affine group
  • Euclidean group
  • Gauss curvature
  • Mean curvature
  • Moving frame
  • Pick invariant


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