In the
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
of
surface
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
s, a Darboux frame is a natural
moving frame
In mathematics, a moving frame is a flexible generalization of the notion of an ordered basis of a vector space often used to study the extrinsic differential geometry of smooth manifolds embedded in a homogeneous space.
Introduction
In lay te ...
constructed on a surface. It is the analog of the
Frenet–Serret frame as applied to surface geometry. A Darboux frame exists at any non-
umbilic
In the differential geometry of surfaces in three dimensions, umbilics or umbilical points are points on a surface that are locally spherical. At such points the normal curvatures in all directions are equal, hence, both principal curvatures are e ...
point of a surface embedded in
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
. It is named after French mathematician
Jean Gaston Darboux
Jean-Gaston Darboux FAS MIF FRS FRSE (14 August 1842 – 23 February 1917) was a French mathematician.
Life
According this birth certificate he was born in Nîmes in France on 14 August 1842, at 1 am. However, probably due to the midni ...
.
Darboux frame of an embedded curve
Let ''S'' be an oriented surface in three-dimensional Euclidean space E
3. 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.
Definition
At each point of an oriented surface, one may attach a
unit
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or beat) in a theatrical presentation
Music
* ''Unit'' (alb ...
normal vector in a unique way, as soon as an orientation has been chosen for the normal at any particular fixed point. If is a curve in , parametrized by arc length, then the Darboux frame of is defined by
:
(the ''unit tangent'')
:
(the ''unit normal'')
:
(the ''tangent normal'')
The triple defines a
positively oriented orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For examp ...
attached to each point of the curve: a natural moving frame along the embedded curve.
Geodesic curvature, normal curvature, and relative torsion
![Darboux frame](https://upload.wikimedia.org/wikipedia/commons/5/56/Darboux_frame.png)
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
*
(the ''unit tangent'', as above)
*
(the ''Frenet normal vector'')
*
(the ''Frenet 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 of N and B produces the pair t and u:
:
Taking a differential, and applying the
Frenet–Serret formulas yields
:
where:
* κ
''g'' is the
geodesic curvature In Riemannian geometry, the geodesic curvature k_g of a curve \gamma measures how far the curve is from being a geodesic. For example, for 1D curves on a 2D surface embedded in 3D space, it is the curvature of the curve projected onto the surface's ...
of the curve,
* κ
''n'' is the normal curvature of the curve, and
* τ
''r'' is the relative torsion (also called geodesic 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 a
principal curve
Principal may refer to:
Title or rank
* Principal (academia), the chief executive of a university
** Principal (education), the office holder/ or boss in any school
* Principal (civil service) or principal officer, the senior management level i ...
of the surface (a ''line of curvature''). In that case, since the principal curves are canonically associated to a surface at all non-
umbilic
In the differential geometry of surfaces in three dimensions, umbilics or umbilical points are points on a surface that are locally spherical. At such points the normal curvatures in all directions are equal, hence, both principal curvatures are e ...
points, the Darboux frame is a canonical
moving frame
In mathematics, a moving frame is a flexible generalization of the notion of an ordered basis of a vector space often used to study the extrinsic differential geometry of smooth manifolds embedded in a homogeneous space.
Introduction
In lay te ...
.
The trihedron
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
1, e
2, and e
3 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.
A trihedron is said to be adapted to a surface if P always lies on the surface and e
3 is the oriented unit normal to the surface at P. In the case of the Darboux frame along an embedded curve, the quadruple
: (P(''s'') = γ(''s''), e
1(''s'') = T(''s''), e
2(''s'') = t(''s''), e
3(''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
:
Change of frame
Suppose that any other adapted trihedron
:(P, e
1, e
2, e
3)
is given for the embedded curve. Since, by definition, P remains the same point on the curve as for the Darboux trihedron, and e
3 = 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 (ω
i,ω
ij) are functions of ''s'', satisfying
:
Structure equations
The
Poincaré lemma In mathematics, especially vector calculus and differential topology, a closed form is a differential form ''α'' whose exterior derivative is zero (), and an exact form is a differential form, ''α'', that is the exterior derivative of another ...
, applied to each double differential ddP, dde
''i'', yields the following
Cartan structure equations. From ddP = 0,
:
From dde
i = 0,
:
The latter are the
Gauss–Codazzi equations
In Riemannian geometry and pseudo-Riemannian geometry, the Gauss–Codazzi equations (also called the Gauss–Codazzi–Weingarten-Mainardi equations or Gauss–Peterson–Codazzi formulas) are fundamental formulas which link together the induced ...
for the surface, expressed in the language of differential forms.
Principal curves
Consider the
second fundamental form
In differential geometry, the second fundamental form (or shape tensor) is a quadratic form on the tangent plane of a smooth surface in the three-dimensional Euclidean space, usually denoted by \mathrm (read "two"). Together with the first fundame ...
of ''S''. This is the symmetric 2-form on ''S'' given by
:
By the
spectral theorem
In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful ...
, there is some choice of frame (e
i) in which (''ii''
ij) is a
diagonal matrix
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal m ...
. The
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
s are the
principal curvatures of the surface. A diagonalizing frame a
1, a
2, a
3 consists of the normal vector a
3, and two principal directions a
1 and a
2. 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
In the differential geometry of surfaces in three dimensions, umbilics or umbilical points are points on a surface that are locally spherical. At such points the normal curvatures in all directions are equal, hence, both principal curvatures are e ...
s of the surface.
Moving frames
The Darboux frame is an example of a natural
moving frame
In mathematics, a moving frame is a flexible generalization of the notion of an ordered basis of a vector space often used to study the extrinsic differential geometry of smooth manifolds embedded in a homogeneous space.
Introduction
In lay te ...
defined on a surface. With slight modifications, the notion of a moving frame can be generalized to a
hypersurface
In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclidea ...
in an ''n''-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
, or indeed any embedded
submanifold
In mathematics, a submanifold of a manifold ''M'' is a subset ''S'' which itself has the structure of a manifold, and for which the inclusion map satisfies certain properties. There are different types of submanifolds depending on exactly which ...
. This generalization is among the many contributions of
Élie Cartan
Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometr ...
to the method of moving frames.
Frames on Euclidean space
A (Euclidean) frame on the Euclidean space E
''n'' is a higher-dimensional analog of the trihedron. It is defined to be an (''n'' + 1)-tuple of vectors drawn from E
''n'', (''v''; ''f''
1, ..., ''f''
''n''), where:
* ''v'' is a choice of
origin
Origin(s) or The Origin may refer to:
Arts, entertainment, and media
Comics and manga
* ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002
* ''The Origin'' (Buffy comic), a 1999 ''Buffy the Vampire Sl ...
of E
''n'', and
* (''f''
1, ..., ''f''
''n'') is an
orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For examp ...
of the vector space based at ''v''.
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
:
where ''A'' is an
orthogonal transformation In linear algebra, an orthogonal transformation is a linear transformation ''T'' : ''V'' → ''V'' on a real inner product space ''V'', that preserves the inner product. That is, for each pair of elements of ''V'', we h ...
and ''x''
0 is a translation. Then, on a frame,
:
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
effective and transitive group action, so ''F''(''n'') is a
principal homogeneous space
In mathematics, a principal homogeneous space, or torsor, for a group ''G'' is a homogeneous space ''X'' for ''G'' in which the stabilizer subgroup of every point is trivial. Equivalently, a principal homogeneous space for a group ''G'' is a non-e ...
of Euc(''n'').
Structure equations
Define the following system of functions ''F''(''n'') → E
''n'':
:
The projection operator ''P'' is of special significance. The inverse image of a point ''P''
−1(''v'') consists of all orthonormal bases with basepoint at ''v''. In particular, ''P'' : ''F''(''n'') → E
''n'' presents ''F''(''n'') as a
principal bundle
In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equi ...
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
''n''.)
The
exterior derivative of ''P'' (regarded as a
vector-valued differential form In mathematics, a vector-valued differential form on a manifold ''M'' is a differential form on ''M'' with values in a vector space ''V''. More generally, it is a differential form with values in some vector bundle ''E'' over ''M''. Ordinary differe ...
) decomposes uniquely as
:
for some system of scalar valued
one-form
In differential geometry, a one-form on a differentiable manifold is a smooth section of the cotangent bundle. Equivalently, a one-form on a manifold M is a smooth mapping of the total space of the tangent bundle of M to \R whose restriction to e ...
s ω
i. Similarly, there is an ''n'' × ''n''
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
of one-forms (ω
ij) such that
:
Since the ''e''
i are orthonormal under the
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
of Euclidean space, the matrix of 1-forms ω
ij is
skew-symmetric. In particular it is determined uniquely by its upper-triangular part (ω
''j''''i'' , ''i'' < ''j''). The system of ''n''(''n'' + 1)/2 one-forms (ω
i, ω
''j''''i'' (''i''<''j'')) gives an
absolute parallelism
In mathematics, a differentiable manifold M of dimension ''n'' is called parallelizable if there exist smooth vector fields
\
on the manifold, such that at every point p of M the tangent vectors
\
provide a basis of the tangent space at p. Equiva ...
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''
i and rotation matrix (''A''
''j''''i''). Then the following are readily checked by the invariance of the exterior derivative under
pullback
In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward.
Precomposition
Precomposition with a function probably provides the most elementary notion of pullback: i ...
:
:
:
Furthermore, by the
Poincaré lemma In mathematics, especially vector calculus and differential topology, a closed form is a differential form ''α'' whose exterior derivative is zero (), and an exact form is a differential form, ''α'', that is the exterior derivative of another ...
, one has the following structure equations
:
:
Adapted frames and the Gauss–Codazzi equations
Let φ : ''M'' → E
''n'' be an embedding of a ''p''-dimensional
smooth manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
into a Euclidean space. The space of adapted frames on ''M'', denoted here by ''F''
φ(''M'') is the collection of tuples (''x''; ''f''
1,...,''f''
n) where ''x'' ∈ ''M'', and the ''f''
i form an orthonormal basis of E
''n'' such that ''f''
1,...,''f''
''p'' are tangent to φ(''M'') at φ(''v'').
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
i,ω
ji) 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.
In detail, the projection π : ''F''(''M'') → ''M'' given by π(''x''; ''f''
i) = ''x'' gives ''F''(''M'') the structure of a
principal bundle
In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equi ...
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''
''i'') := (φ(''v'');''f''
''i'') ∈ ''F''(''n''). 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 ''f''
1...''f''
p are tangent to ''M'' while the others are normal, the structure equations naturally split into their tangential and normal contributions.
[Though treated by Sternberg (1964), this explicit description is from Spivak (1999) chapters III.1 and IV.7.C.] 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
:
since these forms generate the submanifold φ(''M'') (in the sense of the
Frobenius integration theorem
In mathematics, Frobenius' theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an overdetermined system of first-order homogeneous linear partial differential equations. In modern geometric te ...
.)
The first set of structural equations now becomes
:
Of these, the latter implies by
Cartan's lemma that
:
where ''s''
μab is ''symmetric'' on ''a'' and ''b'' (the
second fundamental form
In differential geometry, the second fundamental form (or shape tensor) is a quadratic form on the tangent plane of a smooth surface in the three-dimensional Euclidean space, usually denoted by \mathrm (read "two"). Together with the first fundame ...
s of φ(''M'')). Hence, equations (1) are the Gauss formulas (see
Gauss–Codazzi equations
In Riemannian geometry and pseudo-Riemannian geometry, the Gauss–Codazzi equations (also called the Gauss–Codazzi–Weingarten-Mainardi equations or Gauss–Peterson–Codazzi formulas) are fundamental formulas which link together the induced ...
). In particular, θ
''b''''a'' is the
connection form In mathematics, and specifically differential geometry, a connection form is a manner of organizing the data of a connection using the language of moving frames and differential forms.
Historically, connection forms were introduced by Élie Carta ...
for the
Levi-Civita connection on ''M''.
The second structural equations also split into the following
:
The first equation is the Gauss equation which expresses the
curvature form In differential geometry, the curvature form describes curvature of a connection on a principal bundle. The Riemann curvature tensor in Riemannian geometry can be considered as a special case.
Definition
Let ''G'' be a Lie group with Lie algeb ...
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.
See also
*
Darboux derivative The Darboux derivative of a map between a manifold and a Lie group is a variant of the standard derivative. It is arguably a more natural generalization of the single-variable derivative. It allows a generalization of the single-variable fundament ...
*
Maurer–Cartan form
In mathematics, the Maurer–Cartan form for a Lie group is a distinguished differential one-form on that carries the basic infinitesimal information about the structure of . It was much used by Élie Cartan as a basic ingredient of his met ...
Notes
References
*
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{{DEFAULTSORT:Darboux Frame
Differential geometry
Differential geometry of surfaces
Curvature (mathematics)