Schwarzian Residue of the Samuelson Obstruction in Tangent Lagrangian 2-Webs

Abstract

The Samuelson condition, a classical area-ratio condition for planar Lagrangian $2$-webs, is equivalent in local web coordinates $(u,v)$ to ${\partial }_u{\partial }_v\log |J_F(u,v)|=0$, where $F$ is the inverse web-coordinate map and $J_F$ is its Jacobian. For the tangent-line family $L_t:y=tx+h(t)$, let $\Psi (u,v)$, for $u\ne v$, be the intersection map of $L_u$ and $L_v$, write $J_{\Psi }$ for its Jacobian, and set $S_h(u,v):={\partial }_u{\partial }_v\log |J_{\Psi }(u,v)|$. Near each diagonal point $(t_0,t_0)$ with $h^{\prime \prime }(t_0)\ne 0$, the obstruction admits the decomposition $S_h(u,v)=1/(v-u{)}^2+R_h(u,v)$, where $R_h$ extends smoothly across the diagonal near $(t_0,t_0)$. We call $R_h(t,t)$ the Schwarzian residue and compute $R_h(t,t)=h^{(4)}(t)/(3h^{\prime \prime }(t))-(4/9)(h^{\prime \prime \prime }(t)/h^{\prime \prime }(t){)}^2=(1/2)\{\sigma ,t\}=-{\kappa }_{\mathrm{aff}}(\sigma )(d\sigma /dt{)}^2$. Here $\{\sigma ,t\}$ denotes the Schwarzian derivative of $\sigma$ with respect to $t$, where $\sigma$ is the equi-affine arclength parameter of the envelope $\gamma (t)=(-h^{\prime }(t),h(t)-t{h}^{\prime }(t))$, and ${\kappa }_{\mathrm{aff}}$ denotes its equi-affine curvature, with the convention ${\gamma }_{\sigma \sigma \sigma }+{\kappa }_{\mathrm{aff}}{\gamma }_{\sigma }=0$. Thus the universal pole accounts for the local failure of the Samuelson condition, while the diagonal finite part carries affine-projective information about the envelope. We also derive the transformation law of the Schwarzian residue under reparametrization of the tangent-line parameter.

13 pages, no figures. Revised the Introduction and Sections 3 and 5 to add the Hess connection and bi-Lagrangian interpretation of the Samuelson obstruction and to clarify and directly derive the equi-affine curvature--Schwarzian identity. Main results unchanged