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| In the [[differential geometry]] of [[surface]]s, a '''Darboux frame''' is a natural [[moving frame]] constructed on a surface. It is the analog of the [[Frenet–Serret formulas|Frenet–Serret frame]] as applied to surface geometry. A Darboux frame exists at any non-[[umbilic]] point of a surface embedded in [[Euclidean space]]. It is named after French mathematician [[Jean Gaston Darboux]].
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| ==Darboux frame of an embedded curve==
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| Let ''S'' be an oriented surface in three-dimensional Euclidean space '''E'''<sup>3</sup>. The construction of Darboux frames on ''S'' first considers frames moving along a curve in ''S'', and then specializes when the curves move in the direction of the [[principal curvatures]].
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| ===Definition===
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| At each point of an oriented surface, one may attach a unit normal '''u''' in a unique way. If γ(''s'') is a curve in ''S'', parametrized by arc length, then the '''Darboux frame''' of γ is defined by
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| : <math> \mathbf{T}(s) = \gamma'(s), </math> (the ''unit tangent'')
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| : <math> \mathbf{u}(s) = \mathbf{u}(\gamma(s)), </math> (the ''unit normal'')
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| : <math> \mathbf{t}(s) = \mathbf{u}(s) \times \mathbf{T}(s), </math> (the ''tangent normal'')
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| The triple '''T''','''t''','''u''' defines a [[positively oriented]][[ orthonormal basis]] attached to each point of the curve: a natural moving frame along the embedded curve.
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| ===Geodesic curvature, normal curvature, and relative torsion===
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| 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
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| : <math> \mathbf{T}(s) = \gamma'(s), </math> (the ''unit tangent'', as above)
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| : <math> \mathbf{N}(s) = \frac{\mathbf{T}'(s)}{\|\mathbf{T}'(s)\|}, </math> (the ''Frenet normal vector'')
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| : <math> \mathbf{B}(s) = \mathbf{T}(s)\times\mathbf{N}(s), </math> (the ''Frenet binormal vector'').
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| Since the tangent vectors are the same in both cases, there is a unique angle α such that a rotation in the plane of '''N''' and '''B''' produces the pair '''t''' and '''u''':
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| :<math>
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| \begin{bmatrix}
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| \mathbf{T}\\
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| \mathbf{t}\\
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| \mathbf{u}
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| \end{bmatrix}
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| =
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| \begin{bmatrix}
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| 1&0&0\\
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| 0&\cos\alpha&\sin\alpha\\
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| 0&-\sin\alpha&\cos\alpha
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| \end{bmatrix}
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| \begin{bmatrix}
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| \mathbf{T}\\
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| \mathbf{N}\\
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| \mathbf{B}
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| \end{bmatrix}.
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| </math>
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| Taking a differential, and applying the [[Frenet–Serret formulas]] yields
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| :<math>
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| \mathrm{d}\begin{bmatrix}
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| \mathbf{T}\\
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| \mathbf{t}\\
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| \mathbf{u}
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| \end{bmatrix}
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| =
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| \begin{bmatrix}
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| 0&\kappa\cos\alpha\, \mathrm{d}s&-\kappa\sin\alpha\, \mathrm{d}s\\
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| -\kappa\cos\alpha\, \mathrm{d}s&0&\tau \, \mathrm{d}s + \mathrm{d}\alpha\\
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| \kappa\sin\alpha\, \mathrm{d}s&-\tau \, \mathrm{d}s - \mathrm{d}\alpha&0
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| \end{bmatrix}
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| \begin{bmatrix}
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| \mathbf{T}\\
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| \mathbf{t}\\
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| \mathbf{u}
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| \end{bmatrix}</math>
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| ::<math>=
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| \begin{bmatrix}
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| 0&\kappa_g \, \mathrm{d}s&\kappa_n \, \mathrm{d}s\\
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| -\kappa_g \, \mathrm{d}s&0&\tau_r \, \mathrm{d}s\\
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| -\kappa_n \, \mathrm{d}s&-\tau_r \, \mathrm{d}s&0
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| \end{bmatrix}
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| \begin{bmatrix}
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| \mathbf{T}\\
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| \mathbf{t}\\
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| \mathbf{u}
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| \end{bmatrix}
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| </math>
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| where:
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| * κ<sub>''g''</sub> is the '''geodesic curvature''' of the curve,
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| * κ<sub>''n''</sub> is the '''normal curvature''' of the curve, and
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| * τ<sub>''r''</sub> is the '''relative torsion''' (also called '''geodesic torsion''') of the curve.
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| ==Darboux frame on a surface==
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| This section specializes the case of the Darboux frame on a curve to the case when the curve is a [[principal curvatures|principal curve]] of the surface (a ''line of curvature''). In that case, since the principal curves are canonically associated to a surface at all non-[[umbilic]] points, the Darboux frame is a canonical [[moving frame]].
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| ===The trihedron===
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| [[Image:Darboux trihedron.svg|thumb|right|A Darboux trihedron consisting of a point '''P''' and three orthonormal vectors '''e'''<sub>1</sub>, '''e'''<sub>2</sub>, '''e'''<sub>3</sub> based at '''P'''.]]
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| The introduction of the trihedron (or ''trièdre''), 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. A '''trihedron''' consists of a point '''P''' in Euclidean space, and three orthonormal vectors '''e'''<sub>1</sub>, '''e'''<sub>2</sub>, and '''e'''<sub>3</sub> based at the point '''P'''. A '''moving trihedron''' is a trihedron whose components depend on one or more parameters. For example, a trihedron moves along a curve if the point '''P''' depends on a single parameter ''s'', and '''P'''(''s'') traces out the curve. Similarly, if '''P'''(''s'',''t'') depends on a pair of parameters, then this traces out a surface.
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| A trihedron is said to be '''adapted to a surface''' if '''P''' always lies on the surface and '''e'''<sub>3</sub> is the oriented unit normal to the surface at '''P'''. In the case of the Darboux frame along an embedded curve, the quadruple
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| : ('''P'''(''s'') = γ(''s''), '''e'''<sub>1</sub>(''s'') = '''T'''(''s''), '''e'''<sub>2</sub>(''s'') = '''t'''(''s''), '''e'''<sub>3</sub>(''s'') = '''u'''(''s''))
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| defines a tetrahedron adapted to the surface into which the curve is embedded.
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| In terms of this trihedron, the structural equations read
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| :<math>
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| \mathrm{d}\begin{bmatrix}
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| \mathbf{P}\\
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| \mathbf{T}\\
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| \mathbf{t}\\
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| \mathbf{u}
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| \end{bmatrix}
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| =
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| \begin{bmatrix}
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| 0&\mathrm{d}s&0&0\\
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| 0&0&\kappa_g \, \mathrm{d}s&k_n \, \mathrm{d}s\\
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| 0&-\kappa_g \, \mathrm{d}s&0&\tau_r \, \mathrm{d}s\\
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| 0&-\kappa_n \, \mathrm{d}s&-\tau_r \, \mathrm{d}s&0
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| \end{bmatrix}
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| \begin{bmatrix}
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| \mathbf{P}\\
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| \mathbf{T}\\
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| \mathbf{t}\\
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| \mathbf{u}
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| \end{bmatrix}.
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| </math>
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| ===Change of frame===
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| Suppose that any other adapted trihedron
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| :('''P''', '''e'''<sub>1</sub>, '''e'''<sub>2</sub>, '''e'''<sub>3</sub>)
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| is given for the embedded curve. Since, by definition, '''P''' remains the same point on the curve as for the Darboux trihedron, and '''e'''<sub>3</sub> = '''u''' is the unit normal, this new trihedron is related to the Darboux trihedron by a rotation of the form
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| :<math>
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| \begin{bmatrix}
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| \mathbf{P}\\
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| \mathbf{e}_1\\
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| \mathbf{e}_2\\
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| \mathbf{e}_3
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| \end{bmatrix}
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| =
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| \begin{bmatrix}
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| 1&0&0&0\\
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| 0&\cos\theta&\sin\theta&0\\
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| 0&-\sin\theta&\cos\theta&0\\
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| 0&0&0&1
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| \end{bmatrix}
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| \begin{bmatrix}
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| \mathbf{P}\\
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| \mathbf{T}\\
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| \mathbf{t}\\
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| \mathbf{u}
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| \end{bmatrix}
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| </math>
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| where θ = θ(''s'') is a function of ''s''. Taking a differential and applying the Darboux equation yields
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| :<math>
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| \begin{align}
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| \mathrm{d}\mathbf{P} & = \mathbf{T} \mathrm{d}s = \omega^1\mathbf{e}_1+\omega^2\mathbf{e}_2\\
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| \mathrm{d}\mathbf{e}_i & = \sum_j \omega^j_i\mathbf{e}_j
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| \end{align}
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| </math>
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| where the (ω<sup>i</sup>,ω<sub>i</sub><sup>j</sup>) are functions of ''s'', satisfying
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| :<math>
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| \begin{align}
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| \omega^1 & = \cos\theta \, \mathrm{d}s,\quad \omega^2 = -\sin\theta \, \mathrm{d}s\\
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| \omega_i^j & = -\omega_j^i\\
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| \omega_1^2 & = \kappa_g \, \mathrm{d}s + \mathrm{d}\theta\\
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| \omega_1^3 & = (\kappa_n\cos\theta + \tau_r\sin\theta) \, \mathrm{d}s\\
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| \omega_2^3 & = -(\kappa_n\sin\theta + \tau_r\cos\theta) \, \mathrm{d}s
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| \end{align}
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| </math>
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| ===Structure equations===
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| The [[Poincaré lemma]], applied to each double differential dd'''P''', dd'''e'''<sub>''i''</sub>, yields the following [[Maurer–Cartan form|Cartan structure equations]]. From dd'''P''' = 0,
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| :<math>
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| \begin{align}
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| \mathrm{d}\omega^1 & =\omega^2\wedge\omega_2^1\\
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| \mathrm{d}\omega^2 & =\omega^1\wedge\omega_1^2\\
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| 0 & =\omega^1\wedge\omega_1^3+\omega^2\wedge\omega_2^3
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| \end{align}
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| </math>
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| From dd'''e'''<sub>i</sub> = 0,
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| :<math>
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| \begin{align}
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| \mathrm{d}\omega_1^2 & =\omega_1^3\wedge\omega_3^2\\
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| \mathrm{d}\omega_1^3 & =\omega_1^2\wedge\omega_2^3\\
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| \mathrm{d}\omega_2^3 & =\omega_2^1\wedge\omega_1^3
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| \end{align}
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| </math>
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| The latter are the [[Gauss–Codazzi equations]] for the surface, expressed in the language of differential forms.
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| ===Principal curves===
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| Consider the [[second fundamental form]] of ''S''. This is the symmetric 2-form on ''S'' given by
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| :<math>
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| II = -\mathrm{d}\mathbf{N}\cdot \mathrm{d}\mathbf{P} = \omega_1^3\odot\omega^1 + \omega_2^3\odot\omega^2
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| =\begin{pmatrix}\omega^1 \omega^2\end{pmatrix}
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| \begin{pmatrix}
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| ii_{11}&ii_{12}\\
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| ii_{21}&ii_{22}
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| \end{pmatrix}
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| \begin{pmatrix}\omega^1\\\omega^2\end{pmatrix}.
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| </math>
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| By the [[spectral theorem]], there is some choice of frame ('''e'''<sub>i</sub>) in which (''ii''<sub>ij</sub>) is a [[diagonal matrix]]. The [[eigenvalue]]s are the [[principal curvatures]] of the surface. A diagonalizing frame '''a'''<sub>1</sub>, '''a'''<sub>2</sub>, '''a'''<sub>3</sub> consists of the normal vector '''a'''<sub>3</sub>, and two principal directions '''a'''<sub>1</sub> and '''a'''<sub>2</sub>. This is called a Darboux frame on the surface. The frame is canonically defined (by an ordering on the eigenvalues, for instance) away from the [[umbilic]]s of the surface.
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| <!-- Still under construction:
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| ==Frames on a ruled surface==
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| Let ''S'' be an oriented [[ruled surface]] in three-dimensional Euclidean space '''E'''<sup>3</sup>. One can, following [[Élie Cartan]],<ref>Cartan (1937), Chapter III.</ref> construct a natural frame on ''S'' analogous to the Darboux frame on an ordinary non-ruled surface.
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| A '''trihedron'''<ref>Cartan uses the term ''trièdre''.</ref> in '''E'''<sup>3</sup> consists of a point ''A'' in '''E'''<sup>3</sup> and a triple of [[orthonormal]] vectors ''I''<sub>1</sub>, ''I''<sub>2</sub>, ''I''<sub>3</sub> based at ''A''. A '''moving trihedron''' is a trihedron depending on a system of parameters.
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| A moving trihedron can be attached to ''S'' as follows.
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| is an assignment of a quadruple of vectors ''A'', ''I''<sub>1</sub>, ''I''<sub>2</sub>, ''I''<sub>3</sub> in '''E'''<sup>3</sup>
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| The first construction applies to a space curve on ''S''. Suppose that
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| -->
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| ==Moving frames==
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| The Darboux frame is an example of a natural [[moving frame]] defined on a surface. 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]]. This generalization is among the many contributions of [[Élie Cartan]] to the method of moving frames.
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| ===Frames on Euclidean space===
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| A (Euclidean) '''frame''' on the Euclidean space '''E'''<sup>''n''</sup> is a higher-dimensional analog of the trihedron. It is defined to be an (''n'' + 1)-tuple of vectors drawn from '''E'''<sup>''n''</sup>, (''v''; ''f''<sub>1</sub>, ..., ''f''<sub>''n''</sub>), where:
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| * ''v'' is a choice of [[origin (mathematics)|origin]] of '''E'''<sup>''n''</sup>, and
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| * (''f''<sub>1</sub>, ..., ''f''<sub>''n''</sub>) is an [[orthonormal basis]] of the vector space based at ''v''.
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| Let ''F''(''n'') be the ensemble of all Euclidean frames. The [[Euclidean group]] acts on ''F''(''n'') as follows. Let φ ∈ Euc(''n'') be an element of the Euclidean group decomposing as
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| :<math>\phi(x) = Ax + x_0</math>
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| where ''A'' is an [[orthogonal transformation]] and ''x''<sub>0</sub> is a translation. Then, on a frame,
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| :<math>\phi(v;f_1,\dots,f_n) := (\phi(v);Af_1, \dots, Af_n).</math>
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| 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 an [[group action|effective and transitive group action]], so ''F''(''n'') is a [[principal homogeneous space]] of Euc(''n'').
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| ===Structure equations===
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| Define the following system of functions ''F''(''n'') → '''E'''<sup>''n''</sup>:<ref>Treatment based on Hermann's Appendix II to Cartan (1983), although he takes this approach for the [[affine group]]. The case of the Euclidean group can be found, in equivalent but slightly more advanced terms, in Sternberg (1967), Chapter VI. Note that we have abused notation slightly (following Hermann and also Cartan) by regarding ''f''<sub>''i''</sub> as elements of the Euclidean space '''E'''<sup>''n''</sup> instead of the vector space '''R'''<sup>''n''</sup> based at ''v''. This subtle distinction does not matter, since ultimately only the differentials of these maps are used.</ref>
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| :<math>\begin{align}
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| P(v; f_1,\dots, f_n) & = v\\
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| e_i(v; f_1,\dots, f_n) & = f_i, \qquad i=1,2,\dots,n.
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| \end{align}
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| </math>
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| The projection operator ''P'' is of special significance. The inverse image of a point ''P''<sup>−1</sup>(''v'') consists of all orthonormal bases with basepoint at ''v''. In particular, ''P'' : ''F''(''n'') → '''E'''<sup>''n''</sup> presents ''F''(''n'') as a [[principal bundle]] whose structure group is the [[orthogonal group]] O(''n''). (In fact this principal bundle is just the tautological bundle of the [[homogeneous space]] ''F''(''n'') → ''F''(''n'')/O(''n'') = '''E'''<sup>''n''</sup>.)
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| The [[exterior derivative]] of ''P'' (regarded as a [[vector-valued differential form]]) decomposes uniquely as
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| :<math>\mathrm{d}P = \sum_i \omega^ie_i,\, </math> | |
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| for some system of scalar valued [[one-form]]s ω<sup>i</sup>. Similarly, there is an ''n'' × ''n'' [[matrix (mathematics)|matrix]] of one-forms (ω<sub>i</sub><sup>j</sup>) such that
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| :<math>\mathrm{d}e_i = \sum_j \omega_i^je_j.</math>
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| Since the ''e''<sub>i</sub> are orthonormal under the [[inner product]] of Euclidean space, the matrix of 1-forms ω<sub>i</sub><sup>j</sup> is [[skew-symmetric]]. In particular it is determined uniquely by its upper-triangular part (ω<sub>''j''</sub><sup>''i''</sup> | ''i'' < ''j''). The system of ''n''(''n'' + 1)/2 one-forms (ω<sup>i</sup>, ω<sub>''j''</sub><sup>''i''</sup> (''i''<''j'')) gives an [[absolute parallelism]] of ''F''(''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 translation ''v''<sup>i</sup> and rotation matrix (''A''<sub>''j''</sup><sup>''i''</sup>). Then the following are readily checked by the invariance of the exterior derivative under [[pullback (differential geometry)|pullback]]:
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| :<math>\phi^*(\omega^i) = (A^{-1})_j^i\omega^j</math>
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| :<math>\phi^*(\omega_j^i) = (A^{-1})_p^i\, \omega_q^p\, A_j^q.</math>
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| Furthermore, by the [[Poincaré lemma]], one has the following '''structure equations'''
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| :<math>\mathrm{d}\omega^i = -\omega_j^i\wedge\omega^j</math>
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| :<math>\mathrm{d}\omega_j^i = -\omega^i_k\wedge\omega^k_j.</math>
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| ===Adapted frames and the Gauss–Codazzi equations===
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| Let φ : ''M'' → '''E'''<sup>''n''</sup> be an embedding of a ''p''-dimensional [[smooth manifold]] into a Euclidean space. The space of '''adapted frames''' on ''M'', denoted here by ''F''<sub>φ</sub>(''M'') is the collection of tuples (''x''; ''f''<sub>1</sup>,...,''f''<sub>n</sub>) where ''x'' ∈ ''M'', and the ''f''<sub>i</sub> form an orthonormal basis of '''E'''<sup>''n''</sup> such that ''f''<sub>1</sub>,...,''f''<sub>''p''</sub> are tangent to φ(''M'') at φ(''v'').<ref>This treatment is from Sternberg (1964) Chapter VI.</ref>
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| Several examples of adapted frames have already been considered. The first vector '''T''' of 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 (ω<sup>i</sup>,ω<sub>j</sub><sup>i</sup>) pullback along φ, and the structural equations are preserved under this pullback. 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.
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| In detail, the projection π : ''F''(''M'') → ''M'' given by π(''x''; ''f''<sub>i</sub>) = ''x'' gives ''F''(''M'') the structure of a [[principal bundle]] on ''M'' (the structure group for the bundle is O(''p'') × O(''n'' − ''p'').) This principal bundle embeds into the bundle of Euclidean frames ''F''(''n'') by φ(''v'';''f''<sub>''i''</sub>) := (φ(''v'');''f''<sub>''i''</sub>) ∈ ''F''(''n''). Hence it is possible to define the pullbacks of the invariant forms from ''F''(''n''):
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| :<math>\theta^i = \phi^*\omega^i,\quad \theta_j^i=\phi^*\omega_j^i.</math>
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| Since the exterior derivative is equivariant under pullbacks, the following structural equations hold
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| :<math>\mathrm{d}\theta^i=-\theta_j^i\wedge\theta^j,\quad \mathrm{d}\theta_j^i = -\theta_k^i\wedge\theta_j^k.</math>
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| Furthermore, because some of the frame vectors ''f''<sub>1</sub>...''f''<sub>p</sub> are tangent to ''M'' while the others are normal, the structure equations naturally split into their tangential and normal contributions.<ref>Though treated by Sternberg (1964), this explicit description is from Spivak (1999) chapters III.1 and IV.7.C.</ref> Let the lowercase Latin indices ''a'',''b'',''c'' range from 1 to ''p'' (i.e., the tangential indices) and the Greek indices μ, γ range from ''p''+1 to ''n'' (i.e., the normal indices). The first observation is that
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| :<math>\theta^\mu = 0,\quad \mu=p+1,\dots,n</math>
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| since these forms generate the submanifold φ(''M'') (in the sense of the [[Frobenius integration theorem]].)
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| The first set of structural equations now becomes
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| :<math>\left.\begin{array}{l}
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| \mathrm{d}\theta^a = -\sum_{b=1}^p\theta_b^a\wedge\theta^b\\
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| \\
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| 0=\mathrm{d}\theta^\mu = -\sum_{b=1}^p \theta_b^\mu\wedge\theta^b
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| \end{array}\right\}\,\,\, (1)
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| </math>
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| Of these, the latter implies by [[Cartan's lemma]] that
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| :<math>
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| \theta_b^\mu = s^\mu_{ab}\theta^a
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| </math>
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| where ''s''<sup>μ</sup><sub>ab</sub> is ''symmetric'' on ''a'' and ''b'' (the [[second fundamental form]]s of φ(''M'')). Hence, equations (1) are the '''Gauss formulas''' (see [[Gauss–Codazzi equations]]). In particular, θ<sub>''b''</sub><sup>''a''</sup> is the [[connection form]] for the [[Levi-Civita connection]] on ''M''.
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| The second structural equations also split into the following
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| :<math>
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| \left.\begin{array}{l}
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| \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^\mu_b\\
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| \\
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| \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\\
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| \\
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| \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
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| \end{array}\right\}\,\,\, (2)
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| </math>
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| The first equation is the '''Gauss equation''' which expresses the [[curvature form]] Ω of ''M'' in terms of the second fundamental form. The second is the '''Codazzi–Mainardi equation''' which expresses the covariant derivatives of the second fundamental form in terms of the normal connection. The third is the '''Ricci equation'''.
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| ==See also==
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| * [[Darboux derivative]]
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| * [[Maurer–Cartan form]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *{{cite book|first=Élie|last=Cartan|year=1937|publisher=Gauthier-Villars|title=La théorie des groupes finis et continus et la géométrie différentielle traitées par la méthode du repère mobile}}
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| *{{cite book|author=Cartan, É (Appendices by Hermann, R.)| title=Geometry of Riemannian spaces|publisher = Math Sci Press, Massachusetts | year = 1983}}
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| * {{cite book|first=Gaston|last=Darboux|authorlink=Gaston Darboux|year=1887,1889,1896|title=Leçons sur la théorie génerale des surfaces: [http://www.hti.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ABV4153.0001.001 Volume I], [http://www.hti.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ABV4153.0002.001 Volume II], [http://www.hti.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ABV4153.0003.001 Volume III], [http://www.hti.umich.edu/cgi/t/text/text-idx?c=umhistmath;idno=ABV4153.0004.001 Volume IV]
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| |publisher=Gauthier-Villars}}
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| * {{cite book|first=Heinrich|last=Guggenheimer|title=Differential Geometry|year=1977|publisher=Dover|chapter=Chapter 10. Surfaces|isbn=0-486-63433-7}}
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| * {{cite book|last=Spivak|first=Michael|title=A Comprehensive introduction to differential geometry (Volume 3)|year=1999|publisher=Publish or Perish|isbn=0-914098-72-1}}
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| * {{cite book|last=Spivak|first=Michael|title=A Comprehensive introduction to differential geometry (Volume 4)|year=1999|publisher=Publish or Perish|isbn=0-914098-73-X}}
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| {{curvature}}
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| {{DEFAULTSORT:Darboux Frame}}
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| [[Category:Differential geometry]]
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| [[Category:Differential geometry of surfaces]]
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| [[Category:Curvature (mathematics)]]
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