Laguerre form

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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.

Definition

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

Consider some surface $\mathrm{\Sigma }\hookrightarrow M$ embedded in a three dimensional Riemannian manifold $M$. On $\mathrm{\Sigma }$, 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$$\chi =(e^1{)}^{2}D{a}_{11}+2e^1{e}^{2}D{a}_{12}+(e^2{)}^{2}D{a}_{22}$$or using Einstein summation notation,$$\chi =e^a\otimes e^b\otimes D{a}_{ab}$$