Differential invariants of Maximally symmetric submanifolds

Peter J Olver

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

Let G be a Lie group acting smoothly on a manifold M. A closed, nonsingular submanifold S ⊂ M is called maximally symmetric if its symmetry subgroup Gs ⊂ G has the maximal possible dimension, namely dim G s=dim S., and hence S=Gs · zo is an orbit of Gs · Maximally symmetric submanifolds are characterized by the property that all their differential invariants are constant. In this paper, we explain how to directly compute the numerical values of the differential invariants of a maximally symmetric submanifold from the infinitesimal generators of its symmetry group. The equivariant method of moving frames is applied to significantly simplify the resulting formulae. The method is illustrated by examples of curves and surfaces in various classical geometries. Mathematics Subject Classification 2000: 22F05, 53A55, 58A20.

Original languageEnglish (US)
Pages (from-to)79-99
Number of pages21
JournalJournal of Lie Theory
Volume19
Issue number1
StatePublished - Aug 10 2009

Keywords

  • Differential invariant
  • Homogeneous space
  • Infinitesimal generator
  • Jet
  • Maximally symmetric
  • Moving frame

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