Darboux frame

A Darboux frame exists at any non-umbilic point of a surface embedded in Euclidean space.

Since, by definition, P remains the same point on the curve as for the Darboux trihedron, and e3 = u is the unit normal, this new trihedron is related to the Darboux trihedron by a rotation of the form where θ = θ(s) is a function of s. Taking a differential and applying the Darboux equation yields where the (ωi,ωij) are functions of s, satisfying The Poincaré lemma, applied to each double differential ddP, ddei, yields the following Cartan structure equations.

Consider the second fundamental form of S. This is the symmetric 2-form on S given by By the spectral theorem, there is some choice of frame (ei) in which (iiij) is a diagonal matrix.

With slight modifications, the notion of a moving frame can be generalized to a hypersurface in an n-dimensional Euclidean space, or indeed any embedded submanifold.

The exterior derivative of P (regarded as a vector-valued differential form) decomposes uniquely as for some system of scalar valued one-forms ωi.

The system of n(n + 1)/2 one-forms (ωi, ωji (i

Then the following are readily checked by the invariance of the exterior derivative under pullback: Furthermore, by the Poincaré lemma, one has the following structure equations Let φ : M → En be an embedding of a p-dimensional smooth manifold into a Euclidean space.

Consequently, the resulting system of forms yields structural information about how M is situated inside Euclidean space.

In the case of the Frenet–Serret frame, the structural equations are precisely the Frenet–Serret formulas, and these serve to classify curves completely up to Euclidean motions.

The general case is analogous: the structural equations for an adapted system of frames classifies arbitrary embedded submanifolds up to a Euclidean motion.

Hence it is possible to define the pullbacks of the invariant forms from F(n): Since the exterior derivative is equivariant under pullbacks, the following structural equations hold Furthermore, because some of the frame vectors f1...fp are tangent to M while the others are normal, the structure equations naturally split into their tangential and normal contributions.

The first set of structural equations now becomes Of these, the latter implies by Cartan's lemma that where sμab is symmetric on a and b (the second fundamental forms of φ(M)).