### Abstract

The algebra of differential invariants of a suitably generic surface S ⊂ R^{3}, 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 language | English (US) |
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Pages (from-to) | 230-239 |

Number of pages | 10 |

Journal | Differential Geometry and its Application |

Volume | 27 |

Issue number | 2 |

DOIs | |

State | Published - Apr 1 2009 |

### Keywords

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

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## Cite this

Olver, P. J. (2009). Differential invariants of surfaces.

*Differential Geometry and its Application*,*27*(2), 230-239. https://doi.org/10.1016/j.difgeo.2008.06.020