Laguerre form

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

Definition

This defintion comes from Cartan, translated into more modern notation.

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