On differential invariants of parabolic surfaces
Volume 559 / 2021
Abstract
We study equivalence classes of local graphed analytic surfaces $\{ u = F(x,y)\}$ in $\mathbb R^3$ under the action of the special affine group ${\sf SA}_3(\mathbb R)$, assuming that their Hessian matrices $\bigl( \begin{smallmatrix} F_{xx} & F_{xy} \\ F_{yx} & F_{yy} \end{smallmatrix} \bigr)$ 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 $F_{xx} \neq 0 \equiv F_{xx} F_{yy} - F_{xy}^2$, we quickly encounter the first relative differential invariant \[ \boldsymbol{S} := \frac{F_{xx}F_{xxy}-F_{xy}F_{xxx}}{F_{xx}^2}. \] A surface $\{ u = F(x,y)\}$ is ${\sf SA}_3(\mathbb R)$-equivalent to a curve $\{ u = F(x)\}$ times $\mathbb R_y$ (a cylinder) if and only if $\boldsymbol{S} \equiv 0$. This branch $\boldsymbol{S} \equiv 0$ amounts to the (well-known) ${\sf A}_2(\mathbb R)$-equivalence problem for planar curves.
In the more interesting branch $\boldsymbol{S} \neq 0$, we find the first absolute differential invariant \[ \boldsymbol{W}:=\frac{F_{xx}^2\,F_{xxxy}-F_{xx}\,F_{xy}\,F_{xxxx}+2\,F_{xy}\,F_{xxx}^2 -2\,F_{xx}\,F_{xxx}\,F_{xxy}}{(F_{xx})^2\,\big(F_{xx}\,F_{xxy}-F_{xy}\,F_{xxx}\big)^{2/3}}. \] When $\boldsymbol{W} \equiv 0$, the surface is conical, and we establish that two differential invariants, $\boldsymbol{X}$ of order $5$ and $\boldsymbol{Y}$ of order $7$, generate the full algebra of differential invariants.
In the thickest branch $\boldsymbol{W} \neq 0\, (\neq \boldsymbol{S})$, we find another differential invariant $\boldsymbol{M}$ of order $5$ whose numerator has $57$ differential monomials, and we show that $\boldsymbol{W}$, $\boldsymbol{W}$ 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.
