Darboux frame

In thedifferential geometryofsurfaces,aDarboux frameis a naturalmoving frameconstructed on a surface. It is the analog of theFrenet–Serret frameas applied to surface geometry. A Darboux frame exists at any non-umbilicpoint of a surface embedded inEuclidean space.It is named after French mathematicianJean Gaston Darboux.

Darboux frame of an embedded curve

LetSbe an oriented surface in three-dimensional Euclidean spaceE³.The construction of Darboux frames onSfirst considers frames moving along a curve inS,and then specializes when the curves move in the direction of theprincipal curvatures.

Definition

At each pointpof an oriented surface, one may attach aunitnormal vectoru(p)in a unique way, as soon as an orientation has been chosen for the normal at any particular fixed point. Ifγ(s)is a curve inS,parametrized by arc length, then theDarboux frameofγis defined by

$\mathbf {T} (s)=\gamma '(s),$(theunit tangent)

$\mathbf {u} (s)=\mathbf {u} (\gamma (s)),$(theunit normal)

$\mathbf {t} (s)=\mathbf {u} (s)\times \mathbf {T} (s),$(thetangent normal)

The tripleT,t,udefines apositively orientedorthonormal basisattached to each point of the curve: a natural moving frame along the embedded curve.

Geodesic curvature, normal curvature, and relative torsion

A curve on a surface. The Frenet–Serret frame: tangent in red, the (Frenet) normal in cyan and binormal in purple. The Darboux frame: the tangent in red, the surface normal in blue, and tangent normal in green. Projections along the surface normal and tangent normal shows plane curves whose curvatures are the geodesic curvature and normal curvature respectively.

Note that a Darboux frame for a curve does not yield a natural moving frame on the surface, since it still depends on an initial choice of tangent vector. To obtain a moving frame on the surface, we first compare the Darboux frame of γ with its Frenet–Serret frame. Let

• $\mathbf {T} (s)=\gamma '(s),$(theunit tangent,as above)
• $\mathbf {N} (s)={\frac {\mathbf {T} '(s)}{\|\mathbf {T} '(s)\|}},$(theFrenet normal vector)
• $\mathbf {B} (s)=\mathbf {T} (s)\times \mathbf {N} (s),$(theFrenet binormal vector).

Since the tangent vectors are the same in both cases, there is a unique angle α such that a rotation in the plane ofNandBproduces the pairtandu:

${\begin{bmatrix}\mathbf {T} \\\mathbf {t} \\\mathbf {u} \end{bmatrix}}={\begin{bmatrix}1&0&0\\0&\cos \ Alpha &\sin \ Alpha \\0&-\sin \ Alpha &\cos \ Alpha \end{bmatrix}}{\begin{bmatrix}\mathbf {T} \\\mathbf {N} \\\mathbf {B} \end{bmatrix}}.$

Taking a differential, and applying theFrenet–Serret formulasyields

${\begin{aligned}\mathrm {d} {\begin{bmatrix}\mathbf {T} \\\mathbf {t} \\\mathbf {u} \end{bmatrix}}&={\begin{bmatrix}0&\kappa \cos \ Alpha \,\mathrm {d} s&-\kappa \sin \ Alpha \,\mathrm {d} s\\-\kappa \cos \ Alpha \,\mathrm {d} s&0&\tau \,\mathrm {d} s+\mathrm {d} \ Alpha \\\kappa \sin \ Alpha \,\mathrm {d} s&-\tau \,\mathrm {d} s-\mathrm {d} \ Alpha &0\end{bmatrix}}{\begin{bmatrix}\mathbf {T} \\\mathbf {t} \\\mathbf {u} \end{bmatrix}}\\&={\begin{bmatrix}0&\kappa _{g}\,\mathrm {d} s&\kappa _{n}\,\mathrm {d} s\\-\kappa _{g}\,\mathrm {d} s&0&\tau _{r}\,\mathrm {d} s\\-\kappa _{n}\,\mathrm {d} s&-\tau _{r}\,\mathrm {d} s&0\end{bmatrix}}{\begin{bmatrix}\mathbf {T} \\\mathbf {t} \\\mathbf {u} \end{bmatrix}}\end{aligned}}$

where:

• κ_gis thegeodesic curvatureof the curve,
• κ_nis thenormal curvatureof the curve, and
• τ_ris therelative torsion(also calledgeodesic torsion) of the curve.

Darboux frame on a surface

This section specializes the case of the Darboux frame on a curve to the case when the curve is aprincipal curveof the surface (aline of curvature). In that case, since the principal curves are canonically associated to a surface at all non-umbilicpoints, the Darboux frame is a canonicalmoving frame.

The trihedron

A Darboux trihedron consisting of a pointPand three orthonormal vectorse₁,e₂,e₃based atP.

The introduction of the trihedron (ortrièdre), an invention of Darboux, allows for a conceptual simplification of the problem of moving frames on curves and surfaces by treating the coordinates of the point on the curve and the frame vectors in a uniform manner. Atrihedronconsists of a pointPin Euclidean space, and three orthonormal vectorse₁,e₂,ande₃based at the pointP.Amoving trihedronis a trihedron whose components depend on one or more parameters. For example, a trihedron moves along a curve if the pointPdepends on a single parameters,andP(s) traces out the curve. Similarly, ifP(s,t) depends on a pair of parameters, then this traces out a surface.

A trihedron is said to beadapted to a surfaceifPalways lies on the surface ande₃is the oriented unit normal to the surface atP.In the case of the Darboux frame along an embedded curve, the quadruple

(P(s) = γ(s),e₁(s) =T(s),e₂(s) =t(s),e₃(s) =u(s))

defines a tetrahedron adapted to the surface into which the curve is embedded.

In terms of this trihedron, the structural equations read

$\mathrm {d} {\begin{bmatrix}\mathbf {P} \\\mathbf {T} \\\mathbf {t} \\\mathbf {u} \end{bmatrix}}={\begin{bmatrix}0&\mathrm {d} s&0&0\\0&0&\kappa _{g}\,\mathrm {d} s&\kappa _{n}\,\mathrm {d} s\\0&-\kappa _{g}\,\mathrm {d} s&0&\tau _{r}\,\mathrm {d} s\\0&-\kappa _{n}\,\mathrm {d} s&-\tau _{r}\,\mathrm {d} s&0\end{bmatrix}}{\begin{bmatrix}\mathbf {P} \\\mathbf {T} \\\mathbf {t} \\\mathbf {u} \end{bmatrix}}.$

Change of frame

Suppose that any other adapted trihedron

(P,e₁,e₂,e₃)

is given for the embedded curve. Since, by definition,Premains the same point on the curve as for the Darboux trihedron, ande₃=uis the unit normal, this new trihedron is related to the Darboux trihedron by a rotation of the form

${\begin{bmatrix}\mathbf {P} \\\mathbf {e} _{1}\\\mathbf {e} _{2}\\\mathbf {e} _{3}\end{bmatrix}}={\begin{bmatrix}1&0&0&0\\0&\cos \theta &\sin \theta &0\\0&-\sin \theta &\cos \theta &0\\0&0&0&1\end{bmatrix}}{\begin{bmatrix}\mathbf {P} \\\mathbf {T} \\\mathbf {t} \\\mathbf {u} \end{bmatrix}}$

where θ = θ(s) is a function ofs.Taking a differential and applying the Darboux equation yields

${\begin{aligned}\mathrm {d} \mathbf {P} &=\mathbf {T} \mathrm {d} s=\ Omega ^{1}\mathbf {e} _{1}+\ Omega ^{2}\mathbf {e} _{2}\\\mathrm {d} \mathbf {e} _{i}&=\sum _{j}\ Omega _{i}^{j}\mathbf {e} _{j}\end{aligned}}$

where the (ωⁱ,ω_i^j) are functions ofs,satisfying

${\begin{aligned}\ Omega ^{1}&=\cos \theta \,\mathrm {d} s,\quad \ Omega ^{2}=-\sin \theta \,\mathrm {d} s\\\ Omega _{i}^{j}&=-\ Omega _{j}^{i}\\\ Omega _{1}^{2}&=\kappa _{g}\,\mathrm {d} s+\mathrm {d} \theta \\\ Omega _{1}^{3}&=(\kappa _{n}\cos \theta +\tau _{r}\sin \theta )\,\mathrm {d} s\\\ Omega _{2}^{3}&=-(\kappa _{n}\sin \theta +\tau _{r}\cos \theta )\,\mathrm {d} s\end{aligned}}$

Structure equations

ThePoincaré lemma,applied to each double differential ddP,dde_i,yields the followingCartan structure equations.From ddP= 0,

${\begin{aligned}\mathrm {d} \ Omega ^{1}&=\ Omega ^{2}\wedge \ Omega _{2}^{1}\\\mathrm {d} \ Omega ^{2}&=\ Omega ^{1}\wedge \ Omega _{1}^{2}\\0&=\ Omega ^{1}\wedge \ Omega _{1}^{3}+\ Omega ^{2}\wedge \ Omega _{2}^{3}\end{aligned}}$

From dde_i= 0,

${\begin{aligned}\mathrm {d} \ Omega _{1}^{2}&=\ Omega _{1}^{3}\wedge \ Omega _{3}^{2}\\\mathrm {d} \ Omega _{1}^{3}&=\ Omega _{1}^{2}\wedge \ Omega _{2}^{3}\\\mathrm {d} \ Omega _{2}^{3}&=\ Omega _{2}^{1}\wedge \ Omega _{1}^{3}\end{aligned}}$

The latter are theGauss–Codazzi equationsfor the surface, expressed in the language of differential forms.

Principal curves

Consider thesecond fundamental formofS.This is the symmetric 2-form onSgiven by

$\mathrm {I\!I} =-\mathrm {d} \mathbf {N} \cdot \mathrm {d} \mathbf {P} =\ Omega _{1}^{3}\odot \ Omega ^{1}+\ Omega _{2}^{3}\odot \ Omega ^{2}={\begin{pmatrix}\ Omega ^{1}&\ Omega ^{2}\end{pmatrix}}{\begin{pmatrix}ii_{11}&ii_{12}\\ii_{21}&ii_{22}\end{pmatrix}}{\begin{pmatrix}\ Omega ^{1}\\\ Omega ^{2}\end{pmatrix}}.$

By thespectral theorem,there is some choice of frame (e_i) in which (ii_ij) is adiagonal matrix.Theeigenvaluesare theprincipal curvaturesof the surface. A diagonalizing framea₁,a₂,a₃consists of the normal vectora₃,and two principal directionsa₁anda₂.This is called a Darboux frame on the surface. The frame is canonically defined (by an ordering on the eigenvalues, for instance) away from theumbilicsof the surface.

Moving frames

The Darboux frame is an example of a naturalmoving framedefined on a surface. With slight modifications, the notion of a moving frame can be generalized to ahypersurfacein ann-dimensionalEuclidean space,or indeed any embeddedsubmanifold.This generalization is among the many contributions ofÉlie Cartanto the method of moving frames.

Frames on Euclidean space

A (Euclidean)frameon the Euclidean spaceEⁿis a higher-dimensional analog of the trihedron. It is defined to be an (n+ 1)-tuple of vectors drawn fromEⁿ,(v;f₁,...,f_n), where:

• vis a choice oforiginofEⁿ,and
• (f₁,...,f_n) is anorthonormal basisof the vector space based atv.

LetF(n) be the ensemble of all Euclidean frames. TheEuclidean groupacts onF(n) as follows. Let φ ∈ Euc(n) be an element of the Euclidean group decomposing as

$\phi (x)=Ax+x_{0}$

whereAis anorthogonal transformationandx₀is a translation. Then, on a frame,

$\phi (v;f_{1},\dots,f_{n}):=(\phi (v);Af_{1},\dots,Af_{n}).$

Geometrically, the affine group moves the origin in the usual way, and it acts via a rotation on the orthogonal basis vectors since these are "attached" to the particular choice of origin. This is aneffective and transitive group action,soF(n) is aprincipal homogeneous spaceof Euc(n).

Structure equations

Define the following system of functionsF(n) →Eⁿ:

${\begin{aligned}P(v;f_{1},\dots,f_{n})&=v\\e_{i}(v;f_{1},\dots,f_{n})&=f_{i},\qquad i=1,2,\dots,n.\end{aligned}}$

The projection operatorPis of special significance. The inverse image of a pointP⁻¹(v) consists of all orthonormal bases with basepoint atv.In particular,P:F(n) →EⁿpresentsF(n) as aprincipal bundlewhose structure group is theorthogonal groupO(n). (In fact this principal bundle is just the tautological bundle of thehomogeneous spaceF(n) →F(n)/O(n) =Eⁿ.)

Theexterior derivativeofP(regarded as avector-valued differential form) decomposes uniquely as

$\mathrm {d} P=\sum _{i}\ Omega ^{i}e_{i},\,$

for some system of scalar valuedone-formsωⁱ.Similarly, there is ann×nmatrixof one-forms (ω_i^j) such that

$\mathrm {d} e_{i}=\sum _{j}\ Omega _{i}^{j}e_{j}.$

Since thee_iare orthonormal under theinner productof Euclidean space, the matrix of 1-forms ω_i^jisskew-symmetric.In particular it is determined uniquely by its upper-triangular part (ω_jⁱ|i<j). The system ofn(n+ 1)/2 one-forms (ωⁱ,ω_jⁱ(i<j)) gives anabsolute parallelismofF(n), since the coordinate differentials can each be expressed in terms of them. Under the action of the Euclidean group, these forms transform as follows. Let φ be the Euclidean transformation consisting of a translationvⁱand rotation matrix (A_jⁱ). Then the following are readily checked by the invariance of the exterior derivative underpullback:

$\phi ^{*}(\ Omega ^{i})=(A^{-1})_{j}^{i}\ Omega ^{j}$

$\phi ^{*}(\ Omega _{j}^{i})=(A^{-1})_{p}^{i}\,\ Omega _{q}^{p}\,A_{j}^{q}.$

Furthermore, by thePoincaré lemma,one has the followingstructure equations

$\mathrm {d} \ Omega ^{i}=-\ Omega _{j}^{i}\wedge \ Omega ^{j}$

$\mathrm {d} \ Omega _{j}^{i}=-\ Omega _{k}^{i}\wedge \ Omega _{j}^{k}.$

Adapted frames and the Gauss–Codazzi equations

Let φ:M→Eⁿbe an embedding of ap-dimensionalsmooth manifoldinto a Euclidean space. The space ofadapted framesonM,denoted here byF_φ(M) is the collection of tuples (x;f₁,...,f_n) wherex∈M,and thef_iform an orthonormal basis ofEⁿsuch thatf₁,...,f_pare tangent to φ(M) at φ(x).

Several examples of adapted frames have already been considered. The first vectorTof the Frenet–Serret frame (T,N,B) is tangent to a curve, and all three vectors are mutually orthonormal. Similarly, the Darboux frame on a surface is an orthonormal frame whose first two vectors are tangent to the surface. Adapted frames are useful because the invariant forms (ωⁱ,ω_jⁱ) pullback along φ, and the structural equations are preserved under this pullback. Consequently, the resulting system of forms yields structural information about howMis 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.

In detail, the projection π:F(M) →Mgiven by π(x;f_i) =xgivesF(M) the structure of aprincipal bundleonM(the structure group for the bundle is O(p) × O(n−p).) This principal bundle embeds into the bundle of Euclidean framesF(n) by φ(v;f_i):= (φ(v);f_i) ∈F(n). Hence it is possible to define the pullbacks of the invariant forms fromF(n):

$\theta ^{i}=\phi ^{*}\ Omega ^{i},\quad \theta _{j}^{i}=\phi ^{*}\ Omega _{j}^{i}.$

Since the exterior derivative is equivariant under pullbacks, the following structural equations hold

$\mathrm {d} \theta ^{i}=-\theta _{j}^{i}\wedge \theta ^{j},\quad \mathrm {d} \theta _{j}^{i}=-\theta _{k}^{i}\wedge \theta _{j}^{k}.$

Furthermore, because some of the frame vectorsf₁...f_pare tangent toMwhile the others are normal, the structure equations naturally split into their tangential and normal contributions.Let the lowercase Latin indicesa,b,crange from 1 top(i.e., the tangential indices) and the Greek indices μ, γ range fromp+1 ton(i.e., the normal indices). The first observation is that

$\theta ^{\mu }=0,\quad \mu =p+1,\dots,n$

since these forms generate the submanifold φ(M) (in the sense of theFrobenius integration theorem.)

The first set of structural equations now becomes

$\left.{\begin{array}{l}\mathrm {d} \theta ^{a}=-\sum _{b=1}^{p}\theta _{b}^{a}\wedge \theta ^{b}\\\\0=\mathrm {d} \theta ^{\mu }=-\sum _{b=1}^{p}\theta _{b}^{\mu }\wedge \theta ^{b}\end{array}}\right\}\,\,\,(1)$

Of these, the latter implies byCartan's lemmathat

$\theta _{b}^{\mu }=s_{ab}^{\mu }\theta ^{a}$

wheres^μ_abissymmetriconaandb(thesecond fundamental formsof φ(M)). Hence, equations (1) are theGauss formulas(seeGauss–Codazzi equations). In particular, θ_b^ais theconnection formfor theLevi-Civita connectiononM.

The second structural equations also split into the following

$\left.{\begin{array}{l}\mathrm {d} \theta _{b}^{a}+\sum _{c=1}^{p}\theta _{c}^{a}\wedge \theta _{b}^{c}=\Omega _{b}^{a}=-\sum _{\mu =p+1}^{n}\theta _{\mu }^{a}\wedge \theta _{b}^{\mu }\\\\\mathrm {d} \theta _{b}^{\gamma }=-\sum _{c=1}^{p}\theta _{c}^{\gamma }\wedge \theta _{b}^{c}-\sum _{\mu =p+1}^{n}\theta _{\mu }^{\gamma }\wedge \theta _{b}^{\mu }\\\\\mathrm {d} \theta _{\mu }^{\gamma }=-\sum _{c=1}^{p}\theta _{c}^{\gamma }\wedge \theta _{\mu }^{c}-\sum _{\delta =p+1}^{n}\theta _{\delta }^{\gamma }\wedge \theta _{\mu }^{\delta }\end{array}}\right\}\,\,\,(2)$

The first equation is theGauss equationwhich expresses thecurvature formΩ ofMin terms of the second fundamental form. The second is theCodazzi–Mainardi equationwhich expresses the covariant derivatives of the second fundamental form in terms of the normal connection. The third is theRicci equation.

See also

• Darboux derivative
• Maurer–Cartan form

References

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