Invariants of Surfaces in Three-Dimensional Affine Geometry
Using the method of moving frames, we analyze the algebra of differential invariants for surfaces in three-dimensional affine geometry. For elliptic, hyperbolic, and parabolic points, we show that if the algebra of differential invariants is non-trivial, then it is generically generated by a single...
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| author | Arnaldsson, Örn Valiquette, Francis |
| author_facet | Arnaldsson, Örn Valiquette, Francis |
| citation_txt | Invariants of Surfaces in Three-Dimensional Affine Geometry. Örn Arnaldsson and Francis Valiquette. SIGMA 17 (2021), 033, 25 pages |
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| description | Using the method of moving frames, we analyze the algebra of differential invariants for surfaces in three-dimensional affine geometry. For elliptic, hyperbolic, and parabolic points, we show that if the algebra of differential invariants is non-trivial, then it is generically generated by a single invariant.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 17 (2021), 033, 25 pages
Invariants of Surfaces
in Three-Dimensional Affine Geometry
Örn ARNALDSSON a and Francis VALIQUETTE b
a) Department of Mathematics, University of Iceland, Reykjavik, Ssn. 600169-2039, Iceland
E-mail: ornarnalds@hi.is
b) Department of Mathematics, Monmouth University, West Long Branch, NJ 07764, USA
E-mail: fvalique@monmouth.edu
Received September 03, 2020, in final form March 21, 2021; Published online March 30, 2021
https://doi.org/10.3842/SIGMA.2021.033
Abstract. Using the method of moving frames we analyze the algebra of differential invari-
ants for surfaces in three-dimensional affine geometry. For elliptic, hyperbolic, and parabolic
points, we show that if the algebra of differential invariants is non-trivial, then it is generi-
cally generated by a single invariant.
Key words: affine group; differential invariants; moving frames
2020 Mathematics Subject Classification: 22F05; 53A35; 53A55
1 Introduction
The local geometry of p-dimensional submanifolds S of an m-dimensional manifold M , under
the smooth action of a Lie group G is entirely governed by their differential invariants, in the
sense that two submanifolds are locally congruent if and only if their differential invariants
match [8, 9]. A differential invariant is a (possibly locally defined) smooth function on the
submanifold jet bundle J(∞) = J∞(M,p) that remains unchanged under the prolonged action
of G. This prolonged action on J(∞) splits/reduces to an action on G-invariant subbundles
(called branches of the equivalence problem) whose symmetry properties differ; some branches
having an infinite number of differential invariants of progressively higher and higher order while
others have no invariants. The fundamental basis theorem, first formulated in [16, p. 760], states
that, on branches with non-trivial invariants, all the differential invariants can be generated from
a finite number of low order invariants and their derivatives with respect to p invariant total
derivative operators D1, . . . ,Dp. For example, differential invariants of planar curves under the
special Euclidean group SE(2) can all be expressed in terms of the curvature and its (repeated)
arc-length derivatives [21]. We note that modern proofs of the fundamental basis theorem
can be found in [14, 15, 26] and that this theorem is also frequently called the Lie–Tresse
theorem.
A basic question, then, is to find a minimal generating set of invariants. According to the
above, such a set will completely determine the local geometric properties of submanifolds un-
der G. The equivariant moving frame method is ideally suited for this type of question. Indeed,
the effectiveness of the equivariant moving frame method lies in its recurrence relations, through
which one obtains the complete and explicit structure of the underlying algebra of differential
invariants, and this without requiring explicit coordinate expressions for the moving frame or
the invariants, leading to what is now referred to as the symbolic invariant calculus [17]. In [11]
and [23], this was applied to deduce the surprising result that there is a single generating
invariant for (suitably generic) surfaces in R3 under the projective, conformal, Euclidean and
equi-affine groups. For the Euclidean and equi-affine groups, the algebra of differential invariants
mailto:ornarnalds@hi.is
mailto:fvalique@monmouth.edu
https://doi.org/10.3842/SIGMA.2021.033
2 Ö. Arnaldsson and F. Valiquette
is generically governed by the Gaussian curvature and Pick invariant, respectively. Similarly,
the algebra of differential invariants under the equi-affine group for generic parabolic surfaces
with nonvanishing Pocchiola 4th invariant has recently been shown to be generated by a single
differential invariant in [3].
In the current paper we study the geometry of surfaces under the entire affine group, A(3) =
GL(3) n R3, in detail. We do not restrict ourselves to the most generic branch of surfaces
as in [11, 23], but rather provide all the different branches that have non-trivial invariants.
In each case, we study the algebra of differential invariants and obtain explicit formulas, in terms
of surface jets, for the generating invariants and the invariants responsible for the various bran-
chings. In certain cases, obtaining expressions for the invariants using the direct moving frame
approach proved intractable. We therefore relied on the recently developed technique of recursive
moving frames [24], to obtain the desired coordinate formulas. The main result of our paper is
that whenever a branch admits differential invariants, the differential invariant algebra is (gene-
rically) generated by a single invariant.
It is worth mentioning that, historically, differential geometers have given more attention to
the problem of finding and classifying homogeneous spaces within a given equivalence problem
of submanifolds S ⊂M under the action of a Lie group G. We recall that homogeneous spaces
are, by definition, submanifolds that admit no non-trivial differential invariants, and, in a sense,
the study of these spaces is the “opposite” problem considered in this paper as we focus our
attention to surfaces that admit non-trivial differential invariants. But for completeness, we note
that the classification of homogeneous surfaces in R3 under the equi-affine group can be found in
[10, Theorem 12.4] and [12, Chapter VI]. More recently, normal forms for homogeneous surfaces
in R3 under the general affine group with vanishing equi-affine Pick invariant were found in [1],
and more generally in [5] and [7]. We note that since coordinate expressions for all relative
and differential invariants derived in this paper are known, these could, theoretically, be used
to find normal forms for the homogeneous surfaces. In Section 6 we provide several examples
and show that a more efficient approach to deriving homogeneous surfaces is to integrate the
moving frame equations. Though we emphasize that the study of homogeneous surfaces is not
the main focus of the present paper.
We would be remiss if we failed to acknowledge the classical works of W. Blaschke [2], and
P. Schirokov and A. Schirokov [27] on the subject. Together with [4], and the references therein,
they provide a classical treatment of affine differential geometry. The basic affine differential
invariants can be found in these classical works, and the main contribution of our paper is
the detailed analysis of the structure of the algebra of these differential invariants for surfaces
in affine 3-space.
We note that for parabolic surfaces, the problem studied in this paper is related to the local
geometry of 2-nondegenerate real analytic hypersurfaces S5 ⊂ C3 in CR-geometry [18]. This
correspondence is not considered here, but we note that Question 7.1 in [18, Section 7] is solved
in this paper and corresponds to Case P.1.1 and its subcases. It is worth noting that [18] has
recently been superseded by the work of Doubrov, Merker, and The in [6].
For a summary of the results obtained in this paper we refer the reader to Section 7. As
for the rest of the paper, in Section 2 we recall the notion of a partial moving frame, in-
troduce the recurrence relations that unlock the structure of the algebra of differential in-
variants, summarize the recursive moving frame implementation used to compute coordinate
expressions of invariants, and finally recall basic results pertaining to the algebra of differ-
ential invariants. Sections 3, 4, and 5 contain the main results of this paper. In Section 3
we initiate the normalization process up to order two. At this order there is a splitting ac-
cording to whether points are elliptic, hyperbolic, or parabolic. In Section 4 we simulta-
neously consider elliptic and hyperbolic points. Finally, in Section 5 we consider parabolic
points.
Invariants of Surfaces in Three-Dimensional Affine Geometry 3
2 Background material
In this section we recall basic results pertaining to the method of moving frames. We refer
the reader to the original manuscripts [8, 13, 24] and the book [17] for a more comprehensive
exposition.
2.1 Partial moving frames
In this section we introduce the notion of a partial moving frames as introduced in [24]. Let G
be an r-dimensional Lie group acting on an m-dimensional manifold M . We are interested
with the induced action of G on p-dimensional submanifolds S ⊂ M , where 1 ≤ p < m is
fixed. For 0 ≤ n ≤ ∞, let J(n) = J(n)(M,p) denote the nth order submanifold jet bundle.
Given the local coordinates z = (x, u) =
(
x1, . . . , xp, u1, . . . , uq
)
on M , where x are viewed as
the independent variables and u as the dependent variables, coordinates on J(n) are given by
z(n) =
(
x, u(n)
)
= (. . . , xi, . . . , uαJ , . . . ), where uαJ denote the derivative coordinates of orders
0 ≤ #J ≤ n.
Let S(n) ⊂ J(n) be a G-invariant subbundle of J(n) such that for all g ∈ G near the identity,
g ·S(n) ⊆ S(n). Such an invariant subbundle is specified by a set of invariant differential equations
S(n) =
{
z(n) ∈ J(n) | F
(
z(n)
)
= 0, where F
(
g · z(n)
)∣∣
F (z(n))=0
= 0
}
. (2.1)
The prolongation S(n+1) is obtained by appending the derivatives of the defining equations:
S(n+1) =
{
z(n) ∈ J(n) | F
(
z(n)
)
= 0, (D1F )
(
z(n+1)
)
= 0, . . . , (DpF )
(
z(n+1)
)
= 0
}
,
where Di = Dxi denote the total derivative operators. The induced action of G on S(n) is
called the nth order prolonged action. Borrowing Cartan’s notational convention, we use capital
letters to denote transformed variables: Z(n) = g · z(n). Let B(n) = G × S(n) denote nth order
lifted bundle. For k ≥ n, we introduce the standard projection πkn : B(k) → B(n). The lif-
ted bundle admits a groupoid structure with source map σ(n)
(
g, z(n)
)
= z(n) and target map
Z(n) = τ (n)
(
g, z(n)
)
= g ·z(n) provided by the prolonged action. The action of G on B(n) is given
by right-regularization
Rh
(
g, z(n)
)
=
(
g · h−1, h · z(n)
)
.
Importantly, the target map τ (n)
(
g, z(n)
)
is invariant under the right-regularized action. There-
fore, the pull-back
(
τ (n)
)∗
η of any differential form η on S(n) is invariant on B(n). Since the
cotangent space T ∗B(n) = T ∗G × T ∗S(n) is a direct sum, and G acts separately on its compo-
nents, we may “project” any invariant 1-form on B(n) to an invariant 1-form on S(n). Similarly,
for higher order forms, we have the direct sums∧k
T ∗B(n) =
⊕
i+j=k
(∧i
T ∗G×
∧j
T ∗S(n)
)
,
which the right-regularized action preserves, and so we also have an invariant projection
πJ :
∧k
T ∗B(n) →
∧k
T ∗S(n)
that maps invariant k-forms on B(n) to invariant k-forms on S(n). In practice we apply πJ
by writing a k-form η on B(n) as a direct sum of wedge products of forms on G and S(n) and
then set all T ∗G-terms (which in our case will be the Maurer–Cartan forms) to zero.
4 Ö. Arnaldsson and F. Valiquette
Given a differential form η on S(n), we introduce the lift map
λ(η) := πJ
(
τ (n)
)∗
η, (2.2)
which returns an invariant form on B(n) with only T ∗S(n)-components. The simplest example
is given by the nth order lifted invariants
λ
(
z(n)
)
= g · z(n) = Z(n).
Definition 2.1. A partial right moving frame of order n is a right-invariant local subbundle
ρ̂(n) ⊂ B(n), meaning that Rh
(
ρ̂(n)
)
⊂ ρ̂(n) for all h ∈ G.
In practice, a partial moving frame is obtained by choosing a cross-section K(n) ⊂ S(n)
transversed to the prolonged group action. Then ρ̂(n) =
(
τ (n)
)−1
(K(n)) is a partial moving
frame of order n.
Remark 2.2. We note that as opposed to the standard moving frame definition [8] a partial
moving frame allows for some of the group parameters to not be normalized. More precisely,
if K(n) ⊂ S(n) has codimension kn, then ρ̂(n) also has codimension kn, which implies that r− kn
group parameters remain unnormalized.
Given a partial moving frame ρ̂(n), we introduce the partially normalized invariants
Ẑ(n) =
(
ρ̂(n)
)∗[
λ
(
z(n)
)]
.
The partially normalized invariants are obtained by substituting the normalized group parame-
ters into the lifted invariants Z(n). To simplify the notation in Sections 3, 4, and 5, we do not
include the hat notation over the partially normalized invariants. We hope that the context will
make it clear that we are working with the partially normalized invariants.
2.2 Recurrence relations
The recurrence relations introduced in this section is one of the most important contributions
of [8] to the method of moving frames. These equations unlock the structure of the algebra
of differential invariants (and more generally that of differential forms). One of the key aspects
of these equations is that they can be derived without the coordinate expressions for the (partial)
moving frame, the differential invariants, and the invariant differential forms.
First, a coframe on T ∗B(∞) is given by a basis of Maurer–Cartan forms µ1, . . . , µr, the
horizontal forms dx1, . . . ,dxp, and the basic contact one-forms θαJ = duαJ −uαJ,jdxj . Throughout
this paper we use the Einstein summation convention, where summation occurs over repeated
indices. Since all our computations are performed modulo contact forms, these are omitted from
this point forward.
Applying the lift map (2.2) to the horizontal coframe results in the invariant one-forms
ωi = λ
(
dxi
)
called lifted horizontal forms.
Next, let
vν = ξiν(z)
∂
∂xi
+ φαν (z)
∂
∂uα
, ν = 1, . . . , r = dimG,
Invariants of Surfaces in Three-Dimensional Affine Geometry 5
be a basis of infinitesimal generators dual to the Maurer–Cartan form µ1, . . . , µr. Then the
recurrence relations for the lifted invariants measure the extend to which d ◦ λ 6= λ ◦ d. These
equations are
dXi = ωi + ξiν(Z)µν ,
dUαJ = UαJ,jω
j + φα;Jν
(
Z(#J)
)
µν , (2.3)
where the prolonged vector field coefficients are given by the standard recursive formula
φα;J,jν = Djφ
α;J
ν −
(
Djξ
i
ν
)
· uαJ,i.
Given a partial moving frame ρ̂(n), which we can consider to be in B(∞) using the natural
inclusion i(n) : B(n) ↪→ B(∞), we can then pull-back the lifted recurrence relations (2.3) by ρ̂(n)
to obtain the recurrence relations for the partially normalized invariants
dX̂i = ω̂i + ξiν
(
Ẑ
)
µ̂ν ,
dÛαJ = ÛαJ,jω̂
j + φα;Jν
(
Ẑ(#J)
)
µ̂ν ,
where
ω̂i =
(
ρ̂(n)
)∗
ωi and µ̂ν =
(
ρ̂(n)
)∗
µν
are the partially normalized horizontal one-forms and the partially normalized Maurer–Cartan
forms, respectively.
Remark 2.3. As in the standard moving frame implementation, the symbolic expressions
for the partially normalized Maurer–Cartan forms can be deduced from the recurrence rela-
tions for the phantom invariants, i.e., the lifted invariants that are equal to constant values
by virtue of the moving frame construction. We refer the reader to [8] for more detail.
Remark 2.4. If the prolonged action becomes free on S(n), for a sufficiently large n, we note
that the partial moving frame construction outlined above reproduces the usual moving frame
construction first introduced in [8]. We note that depending on S(n), freeness cannot always be
achieved and this even if the action is locally effective on subsets. Thus, Proposition 9.6 of [8]
holds on regular subsets of the submanifold jet space but not necessarily on invariant subbundles
of the form (2.1). When freeness cannot be attained, the most one can construct is a partial
moving frame.
2.3 Recursive moving frames
For a detailed exposition of the recursive moving frame implementation, we refer the reader to the
original work [24]. One of the main issues of the standard moving frame implementation is that
it first requires computing the prolonged action, which relies on implicit differentiation, and can
lead to unwieldy expressions that limit the method’s practical scope and implementation. This
holds true even when using symbolic softwares such as Mathematica, Maple, or Sage. Some
of the results obtained in this paper are a prime example of this fact. Indeed, we implemented
the standard moving frame machinery in Mathematica and in some cases the software was
unable to solve the normalization equations that produces the moving frame. In those cases we
had to revert to the recursive implementation.
The idea of the recursive moving frame method is, in the spirit of Cartan’s original approach,
to recursively normalize group parameters at a given order before prolonging the action to the
next higher order jet space. Instead of using implicit differentiation to compute the prolonged
6 Ö. Arnaldsson and F. Valiquette
action, the key idea of the recursive moving frame implementation is to use the recurrence
formulas and the expressions for the Maurer–Cartan forms
µ = dg · g−1. (2.4)
To illustrate the recursive moving frame method, assume the prolonged action up to order n is
known and that a partial moving frame ρ̂(n) has been computed using a cross-section K(n) ⊂ S(n).
Assuming, for simplicity, that K(n) is a coordinate cross-section, suppose uαJ = c, with #J = n
is one of the defining equation of K(n). Then ÛαJ = c is a phantom invariant and its recurrence
relation yields
0 = dc = ÛαJ,jω̂
j + φα;Jν
(
Ẑ(n)
)
µ̂ν
so that
ÛαJ,jω̂
j = −φα;Jν
(
Ẑ(n)
)
µ̂ν . (2.5)
By assumption, coordinate expressions for φα;Jν (Ẑ(n)) are known, since the prolonged action
up to order n has been computed, and the partially normalized Maurer–Cartan forms µ̂ν can
be found by substituting the group normalizations into (2.4). Expressing the right-hand side
of (2.5) as a linear combination of the partially normalized horizontal forms ω̂i, we are able to
obtain expressions for the order n+ 1 partially normalized invariants ÛαJ,j .
2.4 The algebra of differential invariants
Assume a moving frame is known or that a partial moving frame has been computed with no
possibility of further group parameter normalizations. Dual to the invariant horizontal forms ωi
are the invariant total derivative operators
Di = Ŵ j
i Di, where
(
Ŵ j
j
)
=
(
ρ̂(n)
)∗(
DjX
i
)−1
. (2.6)
Now, let
dωi = Cijkω
j ∧ ωk mod (unnormalized Maurer–Cartan forms) (2.7)
be the structure equations among the invariant horizontal forms. These equations can be obtai-
ned symbolically by extending the recurrence relations (2.3) to differential forms as done in [13].
Given (2.7), the commutation relations among the invariant total derivative operators are
[Dj ,Dk] = −CijkDi. (2.8)
Fix j, k in (2.8) and apply the commutation relation to p invariants I1, . . . , Ip to obtain
[Dj ,Dk]I` = −CijkDiI`. In matrix form
[Dj ,Dk]I = −DICjk,
where [Dj ,Dk]I = ([Dj ,Dk]I1, . . . , [Dj ,Dk]Ip)T, DI = (DiI`), and Cjk =
(
C1
jk, . . . , C
p
jk
)T
.
If detDI 6≡ 0, then one can solve for Cij
Cjk = −(DI)−1[Dj ,Dk]I, (2.9)
which allows one to express the commutator invariants Cjk in terms of I = (I1, . . . , Ip) and
its invariant derivatives. This is what we refer to as the commutator trick. Notice that given
a single invariant I1, we could have set Ii := D`ikiI1, with 1 ≤ ki ≤ p and `i ≥ 0, in order to write
Invariants of Surfaces in Three-Dimensional Affine Geometry 7
the commutator invariants Cjk as functions of a single invariant and its invariant derivatives.
This observation plays a key role in showing that the algebras of differential invariants for
Euclidean, equi-affine, conformal, and projective surfaces are generically generated by a single
invariant [11, 23, 25]. The commutator trick will also be used in this paper to show that certain
algebras of differential invariants are generated by a single invariant.
We now recall important results about the algebra of differential invariants that can be found
in [8, 22].
Proposition 2.5. The normalized invariants Ẑ(n) provide a complete set of differential invari-
ants of order ≤ n.
By the replacement principle [8, 17], if I
(
z(n)
)
is a differential invariant, then it can be
written in terms of the normalized invariants as I = I
(
Ẑ(n)
)
, which is obtained by replacing the
jet coordinates z(n) by their corresponding normalized invariants Ẑ(n).
Definition 2.6. A set of invariants Igen = {I1, . . . , I`} is said to generate the algebra of diffe-
rential invariants if any differential invariant can be expressed in terms of Igen and its invariant
derivatives (2.6) of any order.
From Proposition 2.5 it follows that if one can show that the normalized invariants Ẑ(∞)
can be written in terms of a set of invariants Igen and its invariant derivatives, then Igen is
a generating set for the algebra of differential invariants.
Theorem 2.7. Given a moving frame ρ̂(n), the normalized invariants Igen =
{
Ẑ(n+1)
}
form
a generating set of differential invariants.
The generating set in Theorem 2.7 is not necessarily minimal. By that we mean that it might
be possible to remove certain non-phantom invariants and still obtain a generating set. To this
day, there is no known result that stipulates how small the generating set can be. But if one
can show that the invariants Igen =
{
Ẑ(n+1)
}
can be expressed in terms of a single invariant I
and its invariant derivatives D1, . . . ,Dp, then the algebra of differential invariants is generated
by a single function. This is the approach used in the following sections to show that the various
differential invariant algebras are generated by a single invariant.
3 Affine action and low-order normalizations
In the following, we consider surfaces S ⊂ R3, which we assume are locally given a graphs
of functions:
S = {z = (x, y, u(x, y))} ⊂ R3.
We are interested in the action of the affine group A(3,R) = GL(3,R) n R3 on these surfaces
given by
Z = Az + b, where A ∈ GL(3,R) and b ∈ R3.
A basis for the algebra of infinitesimal generators is provided by
vxx = x
∂
∂x
, vxy = y
∂
∂x
, vxu = u
∂
∂x
, vyx = x
∂
∂y
, vyy = y
∂
∂y
, vyu = u
∂
∂y
,
vux = x
∂
∂u
, vuy = y
∂
∂u
, vuu = u
∂
∂u
, vx =
∂
∂x
, vy =
∂
∂y
, vu =
∂
∂u
.
8 Ö. Arnaldsson and F. Valiquette
Let
µ =
[
µ ν
0 0
]
with µ =
µxx µxy µxu
µyx µyy µyu
µux µuy µuu
and ν =
µxµy
µu
denote a basis of Maurer–Cartan forms with structure equations
dµ = −µ ∧ µ, dν = −µ ∧ ν.
Then the order zero recurrence relations for the lifted invariants are
dX = ωx +Xµxx + Y µxy + Uµxu + µx,
dY = ωy +Xµyx + Y µyy + Uµyu + µy,
dU = Ujω
j +Xµux + Y µuy + Uµuu + µu,
while for k + ` ≥ 1,
dUXkY ` = UXkY `jω
j − kUXkY `µxx − `UXk+1Y `−1µxy − kUXk−1Y `+1µyx − `UXkY `µyy
+ UXkY `µuu + δ1kδ0`µ
ux + δ0kδ1`µ
uy
−
∑
0≤i≤k
0≤j≤`
(i,j)6=(k,`)
(
k
i
)(
`
j
)[
UXk−iY `−jUXi+1Y jµxu + UXk−iY `−jUXiY j+1µyu
]
,
where there is no summation over k and `, and δij denotes the Kronecker delta function.
Since the action is transitive on J(1), we can set
X = Y = U = UX = UY = 0. (3.1)
In other words, we can choose the cross-section K(1) = {x = y = u = ux = uy = 0} ⊂ J(1).
The recurrence relations for these phantom invariants are
0 = ωx + µx, 0 = ωy + µy, 0 = µu, 0 = UXjω
j + µux, 0 = UY jω
j + µuy.
As mentioned in Section 2.1, from this point onward we omit the use of the hat notation to
denote partially normalized quantities. Solving for the Maurer–Cartan forms yields
µx = −ωx, µy = −ωy, µu = 0, µux = −UXjωj , µuy = −UY jωj . (3.2)
Taking into account the order 0 and 1 normalizations (3.1), and the normalized Maurer–Cartan
forms (3.2), the recurrence relations for the order 2 partially normalized invariants are
dUXX = UXXjω
j + UXX
(
µuu − 2µxx
)
− 2UXY µ
yx,
dUXY = UXY jω
j − UXXµxy + UXY
(
µuu − µxx − µyy
)
− UY Y µyx,
dUY Y = UY Y jω
j + UY Y (µuu − 2µyy)− 2UXY µ
xy. (3.3)
Consider the partially normalized lifted Hessian determinant
H = UXXUY Y − U2
XY .
Since
dH = 2H
(
µuu − µxx − µyy
)
mod (ωx, ωy),
Invariants of Surfaces in Three-Dimensional Affine Geometry 9
we conclude that H is a relative invariant. To obtain an expression for H, we introduce the
determinant
|DX| = det
[
Xx Xy
Yx Yy
]
= det
[
a11 + a13ux a12 + a13uy
a21 + a23uy a22 + a23uy
]
(3.4)
and the Hessian determinant h = uxxuyy − u2xy. Then
H =
a233
|DX|2
h.
Definition 3.1. A point
(
x, y, u(2)
)
of S(2) ∈ J(2) is said to be
� elliptic if h > 0,
� hyperbolic if h < 0,
� parabolic if h = 0.
The remaining analysis depends on the sign of the Hessian determinant. Since most results
for elliptic and hyperbolic points are similar, these two cases are combined together in the next
section. The case of parabolic points is considered in Section 5.
4 Elliptic and hyperbolic points
In this section we work under the assumption that
H = ε = ±1,
with ε = 1 corresponding to the elliptic case and ε = −1 to hyperbolic points. From the
recurrence relations (3.3), we conclude that it is possible to set
UXX = 1, UY Y = ε, UXY = 0. (4.1)
Remark 4.1. In Cartesian coordinates, the normalization equations (4.1) are quadratic in the
group parameters. Therefore, in the process of constructing a moving frame there is a choice
of sign that needs to be made. But since (4.1) holds, no matter the choice made, this does not
affect the algebra of differential invariants of the surface and as such is not important for our
purpose. Thus, as it is customary [25], in the following we omit such ambiguity.
After the normalizations (4.1) have been performed, the recurrence relations for the order 3
partially normalized invariants are
dUX3 = −3µxu − UX3
2
µuu mod (ωx, ωy),
dUX2Y = −εµyu + εUX3µyx − 2UXY 2µyx −
UX2Y
2
µuu mod (ωx, ωy),
dUXY 2 = −εµxu + 2εUX2Y µ
yx − UY 3µyx −
UXY 2
2
µuu mod (ωx, ωy),
dUY 3 = −3µyu + 3εUXY 2µyx −
UY 3
2
µuu mod (ωx, ωy).
Consistent with normalizations performed for elliptic and hyperbolic surfaces in equi-affine geo-
metry [23], we set
UX3 + εUXY 2 = UY 3 + εUX2Y = 0
10 Ö. Arnaldsson and F. Valiquette
and solve for UXY 2 and UX2Y . We are then left with UX3 and UY 3 , whose recurrence relations are
dUX3 = 3εUY 3µyx −
UX3
2
µuu mod (ωx, ωy),
dUY 3 = −3UX3µyx −
UY 3
2
µuu mod (ωx, ωy).
The extent to which one can solve for the partially normalized Maurer–Cartan forms µyx and µuu
depends on the determinant
det
3εUY 3 −UX3
2
−3UX3 −UY 3
2
= −3
2
(
U2
X3 + εU2
Y 3
)
= −3
2
Pε.
We note that Pε is a relative invariant as
dPε = −Pεµuu.
In fact, Pε = P
a33
, where P is the equi-affine Pick invariant
P =
1
16
(
uxxuyy − u2xy
)3 [6uxxuxyuyyuxxxuyyy − 6xxu
2
yyuxxxuxyy − 18uxxuxyuyyuxxyuxyy
+ 12uxxu
2
xyuxxyuyyy − 6u2xxuyyuxxyuyyy + 9uxxu
2
yyu
2
xxy − 6u2xxuxyuxyyuyyy
+ 9u2xxuyyu
2
xyy + u3xxu
2
yyy − 6uxyu
2
yyuxxxuxxy + 12u2xyuyyuxxxuxyy
− 8u3xyuxxxuyyy + u3yyu
2
xxx
]
.
We now need to distinguish the cases where Pε ≡ 0 is identically zero and where Pε 6= 0 does
not vanish. In the elliptic case, we note that if P1 ≡ 0, then UX3 ≡ UY 3 ≡ 0. On the other
hand, in the hyperbolic case, when P−1 ≡ 0, we have that UY 3 ≡ ±UX3 . But, we observe that
under the change of variables (x, y, u) 7→ (x,−y, u), we can always assume that UY 3 = −UX3 .
Therefore, at hyperbolic points there are two cases to consider, either UX3 ≡ 0 or UX3 6= 0.
We combine the different cases as follows:
EH.1 : Pε 6= 0, EH.2 : UX3 ≡ UY 3 ≡ 0, H.3 : UY 3 ≡ −UX3 6= 0.
We note that cases EH.1 and EH.2 hold for both elliptic and hyperbolic points whereas case H.3
is only for hyperbolic points. In local coordinates, since
UXXX =
C1
(
3εa33uxxYx − 4Y 3
x
)
− C2
(
εa33uxx − 4Y 2
x
)√
|h|
√
a33uxx − εY 2
x
4a233u
3
xx|h|3/2
,
UY Y Y =
C1
(
a33uxx − 4εY 2
x
)√
a33uxx − εY 2
x + C2
√
|h|
(
3εa33uxxYx − 4Y 3
x
)
4a233u
3
xx|h|3/2
,
and
Pε =
C2
1 + hC2
2
16a33u3xxh
3
,
where
C1 = 6uxxu
2
xyuxxy − 4u3xyuxxx − 3u2xxuxyuxyy − 3u2xxuyyuxxy + 3uxxuxyuyyuxxx + u3xxuyyy,
C2 = −6uxxuxyuxxy + 4u2xyuxxx + 3u2xxuxyy − uxxuyyuxxx.
the three cases can be restated as
EH.1 : C2
1 + hC2
2 6= 0, EH.2 : C1 ≡ C2 ≡ 0, H.3 : C1 ≡ −C2
√
|h| 6= 0.
Remark 4.2. We remark that the expressions for UXXX and UY Y Y hold provided uxx 6= 0.
From this point forward, we always work on the open dense subset of the jet space where uxx 6= 0.
Invariants of Surfaces in Three-Dimensional Affine Geometry 11
4.1 Case EH.1
When Pε 6= 0, it is possible to set
UX3 = 1, UY 3 = 0.
According to Theorem 2.7, the order 4 differential invariants
UX4 , UX3Y , UX2Y 2 , UXY 3 , UY 4 ,
form a complete set of generating invariants. We now show in fact that the algebra of differential
invariants is generically generated by the single invariant I1 = UY 4 . First, the structure equations
for the invariant coframe ωx, ωy are
dωx =
2ε
3
UXY 3ωx ∧ ωy, dωy =
1
12
(
3UX4 − 6εUX2Y 2 − UY 4
)
ωx ∧ ωy.
Therefore, the Lie bracket of the invariant total derivative operators is
[Dx,Dy] = −2ε
3
UXY 3Dx −
1
12
(
3UX4 − 6εUX2Y 2 − UY 4
)
Dy.
Using the commutator trick (2.9), we can generically solve for I2 = UXY 3 and I3 = UX4−2εUX2Y 2
in terms of I1 and its invariant derivatives. Indeed, applying the commutator trick to I1 and DI1,
where D is a nontrivial invariant total derivative operator, we find that(
−2εI2
3
I1
12 −
I3
4
)
=
(
DxI1 DyI1
DxDI1 DyDI1
)−1(
[Dx,Dy]I1
[Dx,Dy]DI1
)
,
which can be solved for I2 and I3 provided that
DxI1 · DyDI1 −DyI1 · DxDI1 6= 0.
Next, consider the syzygy
DxI1 −DyI2 =
3
2
I3 −
7ε
6
I22 −
1
2
I1I3 +
1
2
I21 +
1
4
(
3U2
X2Y 2 + 6εUX2Y 2 − 2UX3Y UXY 3
)
. (4.2)
This suggests the introduction of the fourth order invariant
I4 = 3U2
X2Y 2 + 6UX2Y 2 − 2UX3Y UXY 3 .
Also, from (4.2) is follows that I4 can be expressed in terms of I1, I2, I3 and their invariant
derivatives. Since I2 and I3 can be expressed in terms of I1 and its invariant derivatives, the
same holds true for I4.
Now, considering the fifth order invariants DiIj , we find, using Mathematica, the syzygy
−216I2DxI2 − 108εI2DyI3 + 36εI2DyI4 + 216I22 − 36I22DxI3 + 12I22DxI4 + 54I1I
2
2
+ 48εI32DxI2 + 24I32DyI3 + 36εI42 − 4εI1I
4
2 − 108I22I3 + 6I1I
2
2I3 − 10εI42I3
− 36εI4DyI2 − 9εI1I4 − 12I2I4DxI2 − 30I22I4 − 2I1I
2
2I4 + 3I22I3I4
+
(
216εDyI2 + 54εI1 + 72I2DxI2 − 432εI2DxI2 − 108I2DyI3 + 36I2DyI4
+ 180I22 + 270εI22 + 12I1I
2
2 + 66εI1I
2
2 − 2I42 − 18I22I3 − 198εI22I3 + 6εI1I
2
2I3
− 27I4 − 36I4DyI2 − 18I1I4 − 33εI22I4
)
UX2Y 2
+
(
162 + 2166DyI2 + 108εDyI2 + 108I1 + 27εI1 − 180I2DxI2 − 144I22 + 198εI22
+ 18I1I
2
2 − 99I22I3 − 54εI4 − 9εI1I4
)
U2
X2Y 2
+
(
81 + 324ε+ 108DyI2 + 54I1 + 54εI1 − 189εI22 − 27I4
)
U3
X2Y 2
+ (162 + 162ε+ 27εI1)U
4
X2Y 2 + 81U5
X2Y 2 = 0.
This is a quintic equation in UX2Y 2 , which can locally be solved in terms of I1, I2, I3, and I4
and their invariant derivatives. This shows the following results.
12 Ö. Arnaldsson and F. Valiquette
Theorem 4.3. If the equi-affine Pick invariant P 6= 0 does not vanish, then the algebra of dif-
ferential invariants is generically generated by the fourth order invariant I1 = UY 4.
Using the method of recursive moving frames, a coordinate expression for the generating
invariant is
UY 4 = 3 +
3
P
√
|h|
(LDyK −KDyL) +
3
4P
Dy
(
PDy(ln |h|)
)
+
3ε
16
(
Dy(ln |h|)
)2
+
3Dy(ln |h|)
4P
√
|h|
(JDyK − IDyL) +
3εDx(ln |h|)
4P
√
|h|
(KDyL− LDyK),
where
Dx =
1
P
√
|h|
(LDx −KDy), Dy =
1
P
√
|h|
(−JDx + IDy)
are invariant total derivative operators and
I =
√
Puxx − εK2, J =
uxy
√
Puxx − εK2 − ε
√
|h|K
uxx
,
L =
uxyK +
√
|h|
√
Puxx − εK2
uxx
, (4.3)
with K a solution to the sextic equation
16εK6 − 24(Puxx)K4 + 9ε(Puxx)2K2 − (Puxx)3C2
1
C2
1 + hC2
2
= 0. (4.4)
Remark 4.4. Over the real numbers, the bi-cubic equation (4.4) has one real solution for K2.
Then, as in Remark 4.1 there is an ambiguity of sign in the definition of K, but this does
not affect the structure of the algebra of differential invariants. Also, on the cross-section,
equation (4.4) reduces to
16εK6 − 24K4 + 9εK2 = 0 (4.5)
so that K2 = 0, 3
4ε . Perturbing (4.5) near the cross-section, the zero root becomes positive,
which implies that K is defined near the cross-section. Finally, we note that on the cross-section
Puxx− εK2 = 1 so that the square roots occurring in (4.3) are well-defined in the neighborhood
of the cross-section.
4.2 Case EH.2
We are now assuming that UX3 ≡ UY 3 ≡ 0. Their recurrence relations imply that
UX4 ≡ 3εUX2Y 2 ≡ UY 4 , UX3Y ≡ UXY 3 ≡ 0. (4.6)
Thus, there is only one fourth order partially normalized invariant. We continue the analysis
using the invariant
UX2Y 2 =
18uxxuxyuxxxuxxy−9u2xxu
2
xxy−(4u2xy+5uxxuyy)u
2
xxx+3uxx(uxxuyy−u2xy)uxxxx
9a33u3xx(uxxuyy − u2xy)
.
Since its recurrence relation is
dUX2Y 2 = −UX2Y 2µuu mod (ωx, ωy),
we now have to consider the cases
EH.2.1 : UX2Y 2 6= 0, EH.2.2 : UX2Y 2 ≡ 0.
Invariants of Surfaces in Three-Dimensional Affine Geometry 13
4.2.1 Case EH.2.1
When UX2Y 2 6= 0, we can normalize
UX2Y 2 = 1.
From (4.6) is follows that all fourth order invariants are constant
UX4 ≡ UY 4 ≡ 3ε, UX2Y 2 = 1, UX3Y ≡ UXY 3 ≡ 0.
Considering their recurrence relations
0 ≡ dUX4 =
(
UX5 − 3εUX3Y 2
)
ωx +
(
UX4Y − 3εUX2Y 3
)
ωy,
0 ≡ dUX3Y = UX4Y ω
x + UX3Y 2ωy,
0 ≡ dUXY 3 = UX2Y 3ωx + UXY 4ωy,
0 ≡ dUY 4 = (UXY 4 − 3εUX3Y 2)ωx +
(
UY 5 − 3εUX2Y 3
)
ωy,
we find that all fifth order invariants vanish. Similarly, the recurrence relations for the fifth
order invariants imply that the sixth order invariants are constant, and so on. Therefore, all
the invariants are constant and there are no further normalizations possible. In particular,
the Maurer–Cartan form µyx cannot be normalized. The structure equations for the coframe
{ωx, ωy, µyx} are
dωx = −εµyx ∧ ωy, dωy = µyx ∧ ωx, dµ = εωy ∧ ωx.
4.2.2 Case EH.2.2
When UX2Y 2 ≡ 0, the same argument as in Case EH.2.1 implies that all higher order partially
normalized invariants vanish. In this case µyx and µuu cannot be normalized and the structure
equations of the coframe {ωx, ωy, µyx, µuu} are
dωx =
1
2
µuu ∧ ωx − εµyx ∧ ωy, dωy = µyx ∧ ωx +
1
2
µuu ∧ ωy, dµyx = 0, dµuu = 0.
4.3 Case H.3
In this section we assume that we are at a hyperbolic point where ε = −1. Also, we are working
under the consideration that UY 3 ≡ −UX3 6= 0. Thus, it is possible to normalize UX3 = 1.
At order 4, the recurrence relation for UY 3 + UX3 ≡ 0, yields the equalities
UXY 3 ≡ −UX4 − 3UX2Y 2 − 3UX3Y , UY 4 ≡ 3UX4 + 6UX2Y 2 + 8UX3Y .
Thus, UX4 , UX3Y , and UX2Y 2 are functionally independent partially normalized invariants.
IntroducingA1
A2
A3
=
1 2 3
1 4 3
1 2 1
UX4
UX3Y
UX2Y 2
,
we have that
dAk = −k
3
Akµ
uu mod (ωx, ωy),
for k = 1, 2, 3. We now need to consider the cases
H.3.1 : A2
1 +A2
2 +A2
3 6= 0, H.3.2 : A1 ≡ A2 ≡ A3 ≡ 0.
14 Ö. Arnaldsson and F. Valiquette
Before considering each case, we note that coordinate expressions for the invariants Ai can
be found using the method of recursive moving frame. We obtained
A1 =
2
√
|h|(2C2hx − hC2,x) + uxy(C2hx − 2hC2,x) + uxx(2hC2,y − C2hy)
3
√
2|h|7/6C4/3
2 a
1/3
33
,
A2 =
3
√
2(uxx(hC2,y − 2C2hy) + (hC2,x − 2C2hx)(
√
|h| − uxy))
uxx|h|11/6(C2a33)2/3
,
A3 =
1
8h3u2xxa33
(√
|h|uxx(3C2hy − 2hC2,y) +
√
|h|uxy(2hC2,x − 3C2hx)
+ h(2hC2,x − 3C2hx) + 4h2hxxuxx − h2yu3xx + 2hxhyu
2
xxuxy
− h2xuxxu2xy − 6hh2xuxx
)
. (4.7)
4.3.1 Case H.3.1
In this case there is Ak, with k ∈ {1, 2, 3}, such that Ak 6= 0. For the sake of the exposition,
assume A3 6= 0. The other possibilities are dealt in a similar fashion. When A3 6= 0, one can
normalize A3 = 1. Then
dA1 =
1
3
[
(3−A1)UX5 + 2(3−A1)UX4Y + (9−A1)UX3Y 2 − 6− 2A2
1 +A1A2 + 12A2
]
ωx
+
1
3
[
(A1−9)UX5 +2(A1−12)U ′X4Y ! +(A1−21)UX3Y 2− 6+2A2
1+A1A2−12A2
]
ωy,
dA2 =
1
6
[
2(3− 2A2)UX5 + 8(3−A2)UX4Y + 2(9− 2A2)UX3Y 2 + 42− 2A1A2 +A2
2
]
ωx
+
1
6
[
2(2A2−9)UX5 +8(A2−6)UX4Y +2(2A2−15)UX3Y 2−42+2A1A2+A2
2
]
ωy,
and we have the structure equations
dωx =
1
12
(8I − 2A1 +A2)ω
x ∧ ωy, dωy =
1
12
(8I − 2A1 −A2)ω
x ∧ ωy,
where I = UX5 + 2UX4Y + UX3Y 2 . Since
DyA2 +DxA2 =
1
3
A2
2 − 2I,
it follows that I can be expressed in terms of A2 and its invariant derivatives. From the syzygy
A1(6I −A2
2) =
A3
2
2
− 6IA2
2 + +3A2(4DxI + 2I + 4DyI − 5DxA2)
+ 6
(
4I2 − 3DxI − 9DyI + 3D2
xA2 + 3DyDxA2
)
,
it follows that A1 can generically be expressed in terms of A2 and its invariant derivatives.
Theorem 4.5. The algebra of differential invariants is generically generated by the single inva-
riant A2.
Remark 4.6. Solving the normalization equation A3 = 1 we obtain
a33 =
1
8h3u2xx
(√
|h|uxx(3C2hy − 2hC2,y) +
√
|h|uxy(2hC2,x − 3C2hx)
+ h(2hC2,x − 3C2hx) + 4h2hxxuxx − h2yu3xx + 2hxhyu
2
xxuxy − h2xuxxu2xy − 6hh2xuxx
)
.
Substituting this group parameter normalization into the formula for A2 in (4.7) yields the
coordinate expression for the generating invariant A2.
Invariants of Surfaces in Three-Dimensional Affine Geometry 15
4.3.2 Case H.3.2
When A1 ≡ A2 ≡ A3 ≡ 0, there is no further group parameter normalizations possible. Then,
the structure equations of the coframe {ωx, ωy, µuu} are
dωx =
1
2
µuu ∧ ωx +
1
6
µuu ∧ ωy, dωy =
1
6
µuu ∧ ωx +
1
2
µuu ∧ ωy, dµuu = 0. (4.8)
5 Parabolic points
At a parabolic point, H = UX2UY 2 − U2
XY ≡ 0. Therefore,
U2
XY ≡ UX2UY 2 (5.1)
and there are now two cases to consider. Namely,
P.1 : UX2 6= 0, P.2 : UX2 ≡ 0.
5.1 Case P.1
When UX2 6= 0, we can solve for UY 2 in (5.1) to obtain
UY 2 ≡
U2
XY
UX2
. (5.2)
Therefore, UX2 and UXY are functionally independent partially normalized invariants. From
the recurrence relations (3.3), we conclude that it is possible to set
UXX = 1, UXY = 0, (5.3)
and (5.2) implies that UY Y ≡ 0. Taking into account the normalizations (5.3) and the equality
UY Y ≡ 0, the recurrence relation for UY Y in (3.3) implies that
UXY 2 ≡ UY 3 ≡ 0, (5.4)
which in turn yields the recurrence relations
0 ≡ dUXY 2 =
(
−2U2
X2Y + UX2Y 2
)
ωx + UXY 3ωy,
0 ≡ dUY 3 = UXY 3ωx + UY 4ωy.
Therefore,
UY 4 ≡ UXY 3 ≡ 0, UX2Y 2 ≡ 2U2
X2Y . (5.5)
Considering the recurrence relations for the third order partially normalized invariants UX3 and
UX2Y , and taking into account the above constraints on the invariants, we find that
dUX3 = −3µxu − 1
2
UX3µuu − 3UX2Y µ
yx mod (ωx, ωy),
dUX2Y = −UX2Y µ
yy mod (ωx, ωy).
From the first equation we conclude that it is possible to normalize UX3 = 0. As for the second
recurrence relation, we have the following cases to consider
P.1.1 : UX2Y 6= 0, P.1.2 : UX2Y ≡ 0.
Introducing the ratio R =
uxy
uxx
, the coordinate expression for UX2Y is
UX2Y =
Rx
Yy −RYx
,
where we recall that Yx and Yy are introduced in (3.4).
16 Ö. Arnaldsson and F. Valiquette
5.1.1 Case P.1.1
When UX2Y 6= 0, we can normalize UX2Y = 1. Then the recurrence relations for the non-
constant fourth order partially normalized lifted invariants, i.e., UX4 and UX3Y , are
dUX4 = −6µyu − UX4µuu − 4UX3Y µ
yx mod (ωx, ωy),
dUX3Y = −1
2
UX3Y µ
uu mod (ωx, ωy). (5.6)
From the first equation, we see that it is possible to normalize UX4 = 0. Next, the exterior
derivative of the constraints (5.5) yields
0 = dUX2Y 2 =
(
UX3Y 2 − 6UX3Y
)
ωx + (UX2Y 3 − 6)ωy,
0 ≡ dUXY 3 =
(
UX2Y 3 − 6
)
ωx + UXY 4ωy,
0 ≡ dUY 4 = UXY 4ωx + UY 5ωy,
from which we obtain the following constraints among the order 5 partially normalized invariants
UY 5 ≡ UXY 4 ≡ 0, UX2Y 3 ≡ 6, UX3Y 2 = 6UX3Y . (5.7)
In light of the second equation in (5.6), we now have to consider the following cases
P.1.1.1 : UX3Y 6= 0, P.1.1.2 : UX3Y ≡ 0,
where
UX3Y =
Rxx
Rx
√
|a33uxx|
.
5.1.2 Case P.1.1.1
When UX3Y 6= 0, we set UX3Y = 1. Then the recurrence relation for UX4Y is
dUX4Y = −2µyx mod (ωx, ωy),
and so we can normalize UX4Y = 0. At this stage, the recurrence relation for the only remaining
fifth order normalized invariant is
dUX5 =
1
3
(
3UX6 + 10UX5Y − 36UX5
)
ωx +
1
9
(
UX5Y + 80− 63UX5
)
ωy. (5.8)
The exterior derivative of the constraints (5.7) yields
0 ≡ dUX3Y 2 =
(
UX4Y 2 − 6
)
ωx +
(
UX3Y 3 − 36
)
ωy,
0 ≡ dUX2Y 3 =
(
UX3Y 3 − 36
)
ωx +
(
UX2Y 4 − 24
)
ωy,
0 ≡ dUXY 4 =
(
UX2Y 4 − 24
)
ωx + UXY 5ωy,
0 ≡ dUY 5 = UXY 5ωx + UY 6ωy.
Thus
UY 6 = 0, UXY 5 = 0, UX2Y 4 = 24, UX3Y 3 = 36, UX4Y 2 = 6,
and it follows that UX6 and UX5Y are the only functionally independent invariants of order 6.
Invariants of Surfaces in Three-Dimensional Affine Geometry 17
From (5.8) we conclude that UX5Y and UX6 can be expressed in terms of UX5 and its invariant
derivatives. It follows from Theorem 2.7 that UX5 generates the algebra of differential invariants.
Introducing the ratios
S =
3uxxuxxxx − 5u2xxx
3u2xx
and L =
uxxx
uxx
,
we have that
UX5 =
Rx
36R4
xx
(
30LRxRxxRxxx − 24LSR2
xRxx − 5L2RxR
2
xx − 60SRxR
2
xx − 40LR3
xx
+ 120R2
xxRxxx − 45RxR
2
xxx + 36R2
xRxxSx
)
.
Finally, the structure equations of the invariant coframe {ωx, ωy} are
dωx = ωy ∧ ωx, dωy =
1
3
ωy ∧ ωx.
5.1.3 Case P.1.1.2
When UX3Y ≡ 0, 0 ≡ dUX3Y = UX4Y ω
x + UX3Y 2ωy, which, when combined with (5.7), implies
that
UY 5 ≡ UXY 4 ≡ UX3Y 2 ≡ UX4Y ≡ 0, UX2Y 3 ≡ 6. (5.9)
Thus, the recurrence relation for the only non-constant order 5 partially normalized invariant,
namely
UX5 =
T
3
√
|a33uxx|
3 ,
where T = 2LS − 3Sx, is
dUX5 = −3
2
UX5µuu mod (ωx, ωy). (5.10)
Next, the recurrence relations for the constant invariants (5.9) are
0 ≡ dUX4Y =
(
UX4Y − 4UX5
)
ωx + UX4Y 2ωy,
0 ≡ dUX3Y 2 = UX4Y 2ωx + UX3Y 3ωy,
0 ≡ dUX2Y 3 = UX3Y 3ωx +
(
UX2Y 4 − 24
)
ωy,
0 ≡ dUXY 4 =
(
UX2Y 4 − 24
)
ωx + UXY 5ωy,
0 ≡ dUY 5 = UXY 5ωx + UY 6ωy.
These equations imply that
UY 6 ≡ UXY 5 ≡ UX3Y 3 ≡ UX4Y 2 ≡ 0, UX2Y 4 ≡ 24, UX5Y ≡ 4UX5 . (5.11)
In light of (5.10), we have the following cases to consider
P.1.1.2.1 : UX5 6= 0, P.1.1.2.2 : UX5 ≡ 0.
18 Ö. Arnaldsson and F. Valiquette
5.1.4 Case P.1.1.2.1
In this case we normalize UX5 = 1. Then the recurrence relation for UX6 is
dUX6 = −3µyx mod (ωx, ωy),
and it is therefore also possible to set UX6 = 0. At this stage all invariants of order 6 or less are
constant and the only non-constant invariant of order 7 is
UX7 = −1
6
− 32/3L2
2T 2/3
− 35/3S
2T 2/3
− 7T 2
x
2 · 31/3T 8/3
+
32/3Txx
T 5/3
.
Similarly, the only non-phantom invariant of order 8 is UX8 . From the recurrence relation
dUX7 =
(
UX8 −
35
2
)
ωx
it follows that UX8 = DxUX7 + 35
2 , and from Theorem 2.7, UX7 generates the algebra of differ-
ential invariants. Finally, the structure equations are
dωx = 0, dωy =
5
3
ωy ∧ ωx.
5.1.5 Case P.1.1.2.2
When UX5 ≡ 0, we have that
0 ≡ dUX5 = UX6ωx,
which when combined with (5.11), implies that all sixth order invariants are constant. Similarly,
all higher order invariants are constant and there are no further possible normalizations. Finally,
the structure equations for the coframe {ωx, ωy, µyx, µuu} are
dωx =
1
2
ωx ∧ ωy, dωy = µyx ∧ ωx,
dµyx =
1
2
µyx ∧ ωy +
1
2
µuu ∧ µyx, dµuu = µyx ∧ ωx.
5.1.6 Case P.1.2
When UX2Y ≡ 0, we have UX2Y ≡ UXY 2 ≡ UY 3 ≡ 0, in light of (5.4). We also recall that UX3
is normalized to zero. From the recurrence relations
0 ≡ dUX2Y = UX3Y ω
x + UX2Y 2ωy,
0 ≡ dUXY 2 = UX2Y 2ωx + UXY 3ωy,
0 ≡ dUY 3 = UXY 3ωx + UY 4ωy,
we conclude that
UXY 3 ≡ UX2Y 2 ≡ UX3Y ≡ UY 4 ≡ 0. (5.12)
Thus,
UX4 =
S
a33uxx
is the lowest order non-zero invariant and the recurrence relation
dUX4 = −UX4µuu mod (ωx, ωy),
leads us to consider the following cases
P.1.2.1 : UX4 6= 0, P.1.2.2 : UX4 ≡ 0.
Invariants of Surfaces in Three-Dimensional Affine Geometry 19
5.1.7 Case P.1.2.1
In this case we can normalize UX4 = 1. Since the recurrence relations for the vanishing invari-
ants (5.12) are of the form 0 ≡ dUJ = UJ,iω
i, all fifth order partially normalized invariants are
zero except for UX5 . Similarly, all sixth order partially normalized invariants are zero except
for UX6 . Since
dUX5 = −1
2
(
10 + 3U2
X5 − 2UX6
)
ωx,
the function UX5 is a genuine differential invariant not depending on the remaining group
parameters. By a similar argument, we see that for k ≥ 5, UXk are genuine differential invariants,
while UXkY ` ≡ 0 for ` > 0 and k + ` ≥ 4. It follows that
UX5 =
3T
S3/2
generates the algebra of differential invariants.
When UX5 = c is constant, it follows that UXk , k ≥ 5, are all constant and the symmetry
group of these surfaces has structure equations
dωx = 0, dωy = µyx ∧ ωx + µyy ∧ ωy,
dµyu = cωx ∧ µyu +
1
3
ωx ∧ µyx + µyu ∧ µyy,
dµyx =
c
2
ωx ∧ µyx + µyu ∧ ωx + µyx ∧ µyy, dµyy = 0.
5.1.8 Case P.1.2.2
If UX4 ≡ 0, then in light of (5.12) all fourth order partially normalized invariants are zero
and there are no non-trivial invariants. These surfaces have a symmetry group with structure
equations
dωx =
1
2
µuu ∧ ωx, dωy = µyx ∧ ωx + µyy ∧ ωy, dµuu = 0, dµyy = 0,
dµyu = µyu ∧ µyy + µuu ∧ µyu, dµyx =
1
2
µuu ∧ µyx + µyx ∧ µyy + µyu ∧ ωx.
5.2 Case P.2
If UX2 ≡ 0, then equation (5.1) implies that UXY ≡ 0. Since
0 = dUXY = −UY Y µyx mod (ωx ∧ ωy),
it follows that UY Y ≡ 0. Such surfaces have a 9-dimensional symmetry group with structure
equations
dωx = µxx ∧ ωx + µxy ∧ ωy, dωy = µyx ∧ ωx + µyy ∧ ωy, dµuu = 0,
dµxx = µyx ∧ µxy, dµxy = µxy ∧ µxx + µyy ∧ µxy,
dµxu = µxu ∧ µxx + µyu ∧ µxy + µuu ∧ µxu, dµyx = µxx ∧ µyx + µyx ∧ µyy,
dµyy = µxy ∧ µyx, dµyu = µxu ∧ µyx + µyu ∧ µyy + µuu ∧ µyu.
20 Ö. Arnaldsson and F. Valiquette
6 Homogeneous surfaces
As mentioned in the introduction, differential geometers have been especially interested in the
study of homogeneous surfaces that arise from the equivalence problem [1, 5, 7]. These sur-
faces are characterized by the property that all relative and differential invariants are constant.
Therefore, homogeneous surfaces are described as solutions to certain systems of partial differ-
ential equations. We now consider several examples, with the understanding that it is not our
intention to recover the full classifications found in [1, 5, 7].
Example 6.1. As our first example, let us consider the branch EH.2.1. Surfaces belonging to
this branch satisfy the partial differential equations UX3 ≡ UY 3 ≡ 0 and the non-degeneracy
condition UX2Y 2 6= 0. In jet coordinates, these conditions translate to the formulas
uxyy =
uxxx
(
uxxuyy − 4u2xy
)
+ 6uxxuxyuxxy
3u2xx
, uyyy =
uyy(3uxxuxxy − 2uxxxuxy)
u2xx
, (6.1)
and (
4u2xxx+ 3uxxuxxxx
)
u2xy − 18uxxuxyuxxxuxxy+ 9u2xxu
2
xxy+ uxxuyy
(
5u2xxx− 3uxxuxxxx
)
6= 0.
Our results say that all surfaces satisfying this system are equivalent, and each is a homogeneous
space with symmetry group of dimension 3. A normal form for this branch can therefore be
taken as any solution to the above system. Completing (6.1) to an involutive system [28], one
obtains a maximally overdetermined fifth order system, which can be solved using the Frobenius
theorem. We find that the non-degenerate quadrics
x2
a2
+
y2
b2
± u2
c2
= 1,
x2
a2
+
y2
b2
− u2
c2
= −1,
with c 6= 0, satisfy the constraints of this branch. These surfaces correspond to case (1) in [1,
Theorem 1.1].
Example 6.2. In Case EH.2.2 we also obtain homogeneous surfaces since all the invariants are
constant. In this case, the surface must be a solution to the system of differential equations
uxyy =
uxxx
(
uxxuyy − 4u2xy
)
+ 6uxxuxyuxxy
3u2xx
, uyyy =
uyy(3uxxuxxy − 2uxxxuxy)
u2xx
,(
4u2xxx+ 3uxxuxxxx
)
u2xy− 18uxxuxyuxxxuxxy+ 9u2xxu
2
xxy+ uxxuyy
(
5u2xxx− 3uxxuxxxx
)
= 0.
One can verify that the remain two non-degenerate quadrics u = y2
b2
± x2
a2
are solutions. Setting
a = b = 1, we recover case (8) of [5, Theorem 1] with α = 2.
Example 6.3. In Case P.1.1.2.2, the homogeneous surface must satisfy the system of differential
equations
uxxuyy − u2xy = 0, 45uxxuxxxuxxxx − 9u2xxuxxxxx − 40u3xxx = 0,
u2xxuxxxy − uxxuxyuxxxx − 2uxxuxxyuxxx + 2uxyu
2
xxx = 0,
and the non-degeneracy conditions
uxx 6= 0, uxxuxxy − uxyuxxx 6= 0.
A solution is given by u = x2y−1, corresponding to case (1) of [5, Theorem 1] with α = 2
and β = −1.
Invariants of Surfaces in Three-Dimensional Affine Geometry 21
Example 6.4. A homogeneous surface in branch P.1.2.2 must satisfy the system of partial
differential equations
uxxuyy − u2xy = 0, uxxyuxx − uxyuxxx = 0, 3uxxuxxxx − 5u2xxx = 0.
A solution is given by u = x2, corresponding to case (1) of [5, Theorem 1].
Example 6.5. A homogeneous surface in branch P.2 will be a solution to the system of diffe-
rential equations
uxx = uxy = uyy = 0.
The general solution being a plane u = ax+ by + c.
The above examples show that attempting to recover the homogeneous surfaces from the sys-
tems of partial differential equations one obtains by setting the relative or differential invariants
to constant values can be extremely challenging as these equations are highly nonlinear and of
high order. Luckily, it is possible to avoid these difficulties by integrating the moving frame
equations instead [10]. To see how this works, let ρ̂(n) =
(
ρ(n), z(n)
)
be a partial right moving
frame. As is customary, we also refer to ρ(n) = ρ(n) ∈ G as a partial right moving frame. Then
let ρ(n) =
(
ρ(n)
)−1
denote the partial left moving frame. Taking the exterior derivative of the
identity ρ(n)ρ(n) = 1, we find that
dρ(n) = −ρ(n)µ∗, (6.2)
where µ∗ =
(
ρ(n)
)∗
µ denotes the right moving frame pull-back of the Maurer–Cartan forms.
To proceed further, let
ρ(n) =
[
E z
0 1
]
,
where E = (e1e2e3) ∈ GL(3) is a frame on the homogeneous surface S ⊂ R3 and z ∈ R3 is
a point in S. It therefore follows that if one can integrate the moving frame equation (6.2)
for ρ(n), a parametrization of the homogeneous surface will be given by the vector z ∈ R3.
We now show how this works with a concrete example.
Example 6.6. In this example we will deduce the homogeneous surface that originates from
Case H.3.2. Recall the structure equations obtained in (4.8). From the third equation, it follows
that, locally,
µuu = da.
Next, introduce
ωx = e−2a/3(ωx + ωy), ωy = e−a/3(ωx − ωy). (6.3)
Using (4.8) we find that
dωx = dωy = 0.
Therefore, locally,
ωx = 2dx, ω = 2dy, (6.4)
22 Ö. Arnaldsson and F. Valiquette
for certain functions x and y. Substituting (6.4) in (6.3) and solving for ωx and ωy we obtain
ωx = e2a/3dx+ ea/3dy, ωy = e2a/3dx− ea/3dy.
Since
µ∗ =
1
2(ωx − ωy) + 1
2µ
uu −1
2(ωx − ωy) + 1
6µ
uu 0 −ωx
1
2(ωx − ωy) + 1
6µ
uu −1
2(ωx − ωy) + 1
2µ
uu 0 −ωy
−ωx ωy µuu 0
0 0 0 0
=
ea/3dy + 1
2da −aa/3dy + 1
6da 0 −e2a/3dx− ea/3dy
ea/3dy + 1
6da −ea/3dy + 1
2da 0 −e2a/3dx+ ea/3dy
−e2a/3dx− ea/3dy e2a/3dx− ea/3dy da 0
0 0 0 0
, (6.5)
equation (6.2) yields
dz =
(
e2a/3dx+ ea/3dy
)
e1 +
(
e2a/3dx− ea/3dy
)
e2,
de1 = −
(
ea/3dy +
da
2
)
e1 −
(
ea/3dy +
da
2
)
e2 +
(
e2a/3dx+ ea/3dy
)
e3,
de2 = −
(
−ea/3dy +
da
6
)
e1 −
(
−ea/3dy +
da
2
)
e2 −
(
e2a/3dx− ea/3dy
)
e3,
de3 = −dae3,
from which we conclude that
zx = e2a/3(e1 + e2), zy = ea/3(e1 − e2),
e1,x = e2a/3e3, e1,y = ea/3(−e1 − e2 + e3), e1,a = −1
2
e1 −
1
6
e2,
e2,x = −e2a/3e3, e2,y = ea/3(e1 + e2 + e3), e2,a = −1
6
e1 −
1
2
e2,
e3,x = e3,y = 0, e3,a = −e3.
and
zxx = 0, zxy = 2eae3, zyy = −2e2a/3(e1 + e2), zyyy = −4eae3.
Integrating the latter system of equations, we obtain
z =
(
e2a/3x+ ea/3y + e2a/3y2
)
e1 +
(
e2a/3x− ea/3y + e2a/3y2
)
e2 − 2ea
(
xy +
1
3
y3
)
e3.
Introducing the variables
x = e2a/3x+ ea/3y + e2a/3y2, y = e2a/3x− ea/3y + e2a/3y2, u = −2ea
(
xy +
1
3
y3
)
,
we find that
u = −x
2
2
+
y2
2
− x3
6
+
x2y
2
− xy2
2
+
y3
6
.
Invariants of Surfaces in Three-Dimensional Affine Geometry 23
Under the change of variables (x, y, u)→ (−x,−y,−u) we get
u =
x2
2
− y2
2
+
x3
6
− x2y
2
+
xy2
2
− y3
6
. (6.6)
This surface is equivalent to the Cayley surface u(x̃, ỹ) = x̃ỹ − 1
3 x̃
3 [19], under the change
of variables x̃ = 1
3√2
(y − x), ỹ = − 1
3√
22
(x+ y).
Finally, one can also verify that (6.6) is a solution to the system of partial differential equations
C1 + C2
√
|h| = A1 = A2 = A3 = 0,
which the surface must satisfy to be in branch H.3.2.
Remark 6.7. We note how the recurrence formula and the equivariant moving frame calculus
facilitated the above computations by providing us with the matrix µ∗ in (6.5) essentially for free.
7 Result summary
In this section we summarize the results obtained in this paper by listing the normal forms
of surfaces, given as graphs of functions u(x, y), for the different, suitably generic, branches of
the equivalence problem we considered in this paper. We also provide the possible dimensions
of the self-symmetry group and recall the branches whose differential invariant algebra is genera-
ted by a single invariant. Note that we do not identify all possible equivalence classes. For homo-
geneous surfaces, this would require a thorough inspection of all possible constant values that
differential invariant can take. For surfaces admitting non-trivial invariants, this would require
a detailed analysis of the signature manifold [20]. Throughout, ε = ±1, with ε = 1 for elliptic
points and ε = −1 for hyperbolic points.
Case EH.1:
u(x, y) =
1
2
x2 + ε
1
2
y2 +
1
6
x3 + ε
1
2
x2y +
∑
i,j≥0
ci(4+j)
1
i!(4 + j)!
xiy4+j +
∑
i+j≥4
j<4
Fij(c)
1
i!j!
xiyj ,
where c is the infinite vector of coefficients ci(4+j), i, j ≥ 0 and Fij are certain universal, deter-
minable, functions thereof. These surfaces have self-symmetry groups of dimension 0, 1 or 2,
depending on the particularities of c. Also, the algebra of differential invariants is generated by
a single fourth order invariant.
Case EH.2.1:
u(x, y) =
1
2
x2 + ε
1
2
y2 +
3ε
4!
x4 +
1
4
x2y2 +
3ε
4!
y4 + h.o.t.,
where h.o.t. are higher order terms. These surfaces have self-symmetry group of dimension 3,
and there are no differential invariants.
Case H.2.2:
u(x, y) =
1
2
x2 + ε
1
2
y2.
These surfaces have self-symmetry groups of dimension 4, and there are no differential inva-
riants.
Case H.3.1:
u(x, y) =
1
2
x2 − 1
2
y2 +
1
6
x3 − 1
2
x2y − ε̃1
2
xy2 + ε̃
1
6
y3
+
∑
i,j≥0
c(2+i)(2+j)
1
(2 + i)!(2 + j)!
x2+iy2+j +
∑
i+j≥4
j<2 or i<2
Fij(c)
1
i!j!
xiyj ,
24 Ö. Arnaldsson and F. Valiquette
where c is the infinite vector of c(2+i)(2+j), i, j ≥ 0, c22 6= 0, and Fij are certain universal,
determinable, functions thereof. These surfaces have self-symmetry groups of dimension 0, 1
or 2, depending on the particularities of c. Furthermore, the algebra of differential invariants is
generated by a single fourth order invariant.
Case H.3.2:
u(x, y) =
1
2
x2 − 1
2
y2 +
1
6
x3 − 1
2
x2y +
1
2
xy2 − 1
6
y3.
The self-symmetry group has dimension 3, and there are no differential invariants.
Case P1.1.1:
u(x, y) =
1
2
x2 +
1
2
x2y +
1
6
x3y +
1
2
x2y2 +
∑
i,j≥0
c(5+i)j
1
(5 + i)!j!
x5+iyj +
∑
i+j≥5
i<5
Fij(c)
1
i!j!
xiyj ,
where c is the infinite vector of c(5+i)(j), i, j ≥ 0 and Fij are certain universal, determinable,
functions thereof. These surfaces have self-symmetry groups of dimension 0, 1 or 2, depending
on the particularities of c. In this case, the algebra of differential invariants is generated by
a fifth order invariant.
Case P.1.1.2.1:
u(x, y) =
1
2
x2 +
1
2
x2y +
1
2
x2y2 +
1
5!
x5 +
1
2
x2y3 +
1
2
x2y4 +
1
30
x5y + h.o.t.,
where h.o.t. are higher order terms. The self-symmetry group has dimension 2, and the diffe-
rential invariant algebra is generated by a seventh order invariant.
Case P.1.1.2.2:
u(x, y) =
1
2
x2 +
1
2
x2y +
1
2
x2y2 +
1
5!
x5 +
1
2
x2y3 +
1
2
x2y4 + h.o.t.,
where h.o.t. are higher order terms. The self-symmetry group has dimension 4, and there are
no differential invariants.
Case P.1.2.1:
u(x, y) =
1
2
x2 +
1
4!
x4 +
∑
i≥0
c(5+i)0
1
(5 + i)!
x5+i.
The self-symmetry group has dimension 3, 4 or 5 depending on the series of c(5+i)0, and the
invariant differential algebra is generated by a fifth invariant.
Case P.1.2.2:
u(x, y) =
1
2
x2.
The symmetry group has dimension 6, and there are no differential invariants.
Case P.2:
u(x, y) = 0
has a 9-dimensional self-symmetry group, and there are no differential invariants.
Acknowledgement
We would like to thank the referees for their valuable comments, which helped improve the
exposition of the paper.
Invariants of Surfaces in Three-Dimensional Affine Geometry 25
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1 Introduction
2 Background material
2.1 Partial moving frames
2.2 Recurrence relations
2.3 Recursive moving frames
2.4 The algebra of differential invariants
3 Affine action and low-order normalizations
4 Elliptic and hyperbolic points
4.1 Case EH.1
4.2 Case EH.2
4.2.1 Case EH.2.1
4.2.2 Case EH.2.2
4.3 Case H.3
4.3.1 Case H.3.1
4.3.2 Case H.3.2
5 Parabolic points
5.1 Case P.1
5.1.1 Case P.1.1
5.1.2 Case P.1.1.1
5.1.3 Case P.1.1.2
5.1.4 Case P.1.1.2.1
5.1.5 Case P.1.1.2.2
5.1.6 Case P.1.2
5.1.7 Case P.1.2.1
5.1.8 Case P.1.2.2
5.2 Case P.2
6 Homogeneous surfaces
7 Result summary
References
|
| id | nasplib_isofts_kiev_ua-123456789-211316 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-15T09:36:07Z |
| publishDate | 2021 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Arnaldsson, Örn Valiquette, Francis 2025-12-29T11:09:43Z 2021 Invariants of Surfaces in Three-Dimensional Affine Geometry. Örn Arnaldsson and Francis Valiquette. SIGMA 17 (2021), 033, 25 pages 1815-0659 2020 Mathematics Subject Classification: 22F05; 53A35; 53A55 arXiv:2009.00670 https://nasplib.isofts.kiev.ua/handle/123456789/211316 https://doi.org/10.3842/SIGMA.2021.033 Using the method of moving frames, we analyze the algebra of differential invariants for surfaces in three-dimensional affine geometry. For elliptic, hyperbolic, and parabolic points, we show that if the algebra of differential invariants is non-trivial, then it is generically generated by a single invariant. We would like to thank the referees for their valuable comments, which helped improve the exposition of the paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Invariants of Surfaces in Three-Dimensional Affine Geometry Article published earlier |
| spellingShingle | Invariants of Surfaces in Three-Dimensional Affine Geometry Arnaldsson, Örn Valiquette, Francis |
| title | Invariants of Surfaces in Three-Dimensional Affine Geometry |
| title_full | Invariants of Surfaces in Three-Dimensional Affine Geometry |
| title_fullStr | Invariants of Surfaces in Three-Dimensional Affine Geometry |
| title_full_unstemmed | Invariants of Surfaces in Three-Dimensional Affine Geometry |
| title_short | Invariants of Surfaces in Three-Dimensional Affine Geometry |
| title_sort | invariants of surfaces in three-dimensional affine geometry |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211316 |
| work_keys_str_mv | AT arnaldssonorn invariantsofsurfacesinthreedimensionalaffinegeometry AT valiquettefrancis invariantsofsurfacesinthreedimensionalaffinegeometry |