Laguerre form

In mathematics, the Laguerre form is a tensor-valued form on an embeded surface, whose ratio with the infinitesimal line element cubed is invariant with respect to a choice of frame.

This defintion comes from Cartan^[1], translated into more modern notation.

Consider some surface ${\displaystyle \Sigma \hookrightarrow M}$ embedded in a three dimensional Riemannian manifold ${\displaystyle M}$. On ${\displaystyle \Sigma }$, define an orthonormal coframe ${\displaystyle e^{a}}$, and let ${\displaystyle a}$ be the second fundamental form, and the exterior covariant derivative ${\displaystyle D}$. The Laguerre form is a tensor-valued form defined by$${\displaystyle \chi =(e^{1})^{2}Da_{11}+2e^{1}e^{2}Da_{12}+(e^{2})^{2}Da_{22}}$$or using Einstein summation notation,$${\displaystyle \chi =e^{a}\otimes e^{b}\otimes Da_{ab}}$$