TY - JOUR

T1 - Differential invariants of surfaces

AU - Olver, Peter J.

PY - 2009/4

Y1 - 2009/4

N2 - 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.

AB - 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.

KW - Differential invariant

KW - Equi-affine group

KW - Euclidean group

KW - Gauss curvature

KW - Mean curvature

KW - Moving frame

KW - Pick invariant

UR - http://www.scopus.com/inward/record.url?scp=61549093454&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=61549093454&partnerID=8YFLogxK

U2 - 10.1016/j.difgeo.2008.06.020

DO - 10.1016/j.difgeo.2008.06.020

M3 - Article

AN - SCOPUS:61549093454

VL - 27

SP - 230

EP - 239

JO - Differential Geometry and its Applications

JF - Differential Geometry and its Applications

SN - 0926-2245

IS - 2

ER -