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Invariants of surfaces in three-dimensional affine geometry. (English) Zbl 1481.53015

The paper performs a detailed analysis of local differential invariants and recurrence relations for surfaces viewed under the action of the affine group. The analysis is not based on any general assumptions and, in particular, includes both the case of hyperbolic surfaces with vanishing Pick invariant and the case of parabolic non-cylindrical surfaces.
The techniques of the paper are mostly straightforward and follow the ideas of [M. Fels and P. J. Olver, Acta Appl. Math. 55, No. 2, 127–208 (1999; Zbl 0937.53013)]. The authors heavily rely on symbolic computations for deriving explicit expressions of differential invariants and recurrence relations.
As a final result, the authors describe:
the branch of elliptic or hyperbolic surfaces with non-vanishing Pick invariant, which has a single generating invariant of order 4;
the branch of hyperbolic surfaces with vanishing Pick invariant, also having a single generator of order 4;
the generic branch of parabolic non-cylindrical surfaces with a single generator of order 5;
a non-generic branch of parabolic non-cylindrical surfaces with a single generator of order 7, which seem to correspond to affine cones;
the branch of cylindrical surfaces with a single generator of order 5;
a number of branches having no differential invariants, which correspond to some specific homogeneous surfaces from the classification [loc. cit.].

MSC:

53A15 Affine differential geometry
22F05 General theory of group and pseudogroup actions
53A55 Differential invariants (local theory), geometric objects

Citations:

Zbl 0937.53013
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References:

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