Dissertationes mathematicae

We study equivalence classes of local graphed analytic surfaces fu = F(x; y)g in R3 under the action of the special affine group SA3(R), assuming that their Hessian matrices Fxx Fxy Fyx Fyy have rank 1 at every point (x; y). Such parabolic surfaces have identically zero Gaussian curvature, hence are developable. After the treatment of the rank 2 case by Olver [Differential Geom. Appl. 27 (2007)], we determine the structures of various algebras of differential invariants in all possible branches, and we employ the power series method in order to compute all incoming relative or absolute differential invariants. Starting with our rank 1 root hypothesis Fxx 6= 0  FxxFyy F2 xy, we quickly encounter the first relative differential invariant A surface fu = F(x; y)g is SA3(R)-equivalent to a curve fu = F(x)g times Ry (a cylinder) if and only if S 0. This branch S 0 amounts to the (well-known) A2(R)-equivalence problem for planar curves. In the more interesting branch S 6= 0, we find the first absolute differential invariant When W 0, the surface is conical, and we establish that two differential invariants, X of order 5 and Y of order 7, generate the full algebra of differential invariants. In the thickest branch W 6= 0 (6= S), we find another differential invariant M of order 5 whose numerator has 57 differential monomials, and we show that M, W are generators. Mainly, we set up the celebrated Fels-Olver recurrence formulas for differential invariants under the assumptions that one or two (relative) differential invariants vanish identically. These degenerate cases, apparently, have not been studied before in the literature, and will be developed further.