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