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 in non-trivial, then it is generically generated by a single invariant.


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, [1,2]. 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 [7, 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 D 1 , . . . , D p . 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, [10]. We note that modern proofs of the Fundamental Basis Theorem can be found in [5,6,15] 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 under G. The equivariant moving frame method is ideally suited to 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, [8]. In [3] and [12], this was applied to deduce the surprising result that there is a single generating invariant for (suitably generic) surfaces in R 3 under the projective, conformal, Euclidean and equi-affine groups. For the Euclidean and equi-affine groups, the algebra of differential invariant is generically governed by the Gaussian curvature and Pick invariant, respectively.
In the current paper we study the geometry of surfaces under the entire affine group, A(3)=GL(3) ⋉ R 3 , in detail. We do not restrict ourselves to the most generic branch of surfaces as in [3,12], but rather provide all the different branches. 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 branchings. 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, [13], 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 (generically) generated by a single invariant.
We note that for parabolic surfaces, the problem studied in this paper is related to the local geometry of 2-nondegenerate real analytic hypersurfaces S 5 ⊂ C 3 in CR-geometry, [9]. This correspondence is not considered here, but we note that Question 7.1 in [9, Sect. 7] is solved in this paper and corresponds to Case P.1.1 and its subcases.
For a summary of the results obtained in this paper we refer the reader to Section 6. As for the rest of the paper, in Section 2 we recall the notion of a partial moving frame, introduce the recurrence relations that unlock the structure of the algebra of differential invariants, summarize the recursive moving frame implementation used to compute coordinate expressions of invariants, and finally recall basic results pertaining to the algebra of differential 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 according to whether points are elliptic, hyperbolic, or parabolic. In Section 4 we simultaneously consider elliptic and hyperbolic points. Finally, in Section 5 we consider parabolic points.

Background Material
In this section we recall basic results pertaining to the method of moving frames. We refer the reader to the original manuscripts [1,4,13] and the book [8] for a more comprehensive exposition.

Partial Moving Frames
In this section we introduce the notion of a partial moving frames as introduced in [13]. 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 n th order submanifold jet bundle. Given the local coordinates z = (x, u) = (x 1 , . . . , x p , u 1 , . . . , u q ) 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) ) = ( . . . x i . . . u α J . . . ), where u α J 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 (2.1) The prolongation S (n+1) is obtained by appending the derivatives of the defining equations: where D i = D x i denote the total derivative operators. The induced action of G on S (n) is called the n th 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 n th order lifted bundle. For k ≥ n, we introduce the standard projection π k n : B (k) → B (n) . The lifted 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 We note that Z (n) is invariant under R h . These quantities are called lifted invariants of order ≤ n, and we introduce the lift map In practice, a partial moving frame is obtained by choosing a cross-section K (n) ⊂ S (n) transversed to the prolonged group action. Then ρ (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, [1], a partial moving frame allows for some of the group parameters to not be normalized. More precisely, if K (n) ⊂ S (n) has codimension k n , then ρ (n) also has codimension k n , which implies that r − k n group parameters remain unnormalized.
Given a partial moving frame ρ (n) , we introduce the partially normalized invariants The partially normalized invariants are obtained by substituting the normalized group parameters 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.

Recurrence relations
The recurrence relations introduced in this section is one of the most important contributions of [1] 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 moving frames, the differential invariants, and the invariant differential forms.
First, a coframe on T * B (∞) is given by a basis of Maurer-Cartan forms µ 1 , . . . , µ r , the horizontal forms dx 1 , . . . , dx p , and the basic contact one-forms θ α J = du α J − u α J,j dx j . Throughout this paper we use 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. Introducing the projection map obtained by setting the Maurer-Cartan forms to zero, the lift map (2.2) extends to the horizontal coframe as follows and the resulting one-forms are called lifted horizontal forms.
be a basis of infinitesimal generators dual to the Maurer-Cartan form µ 1 , . . . , µ r . Then the recurrence relations for the lifted invariants measure the extend to which d • λ = λ • d. These equations are where the prolonged vector field coefficients are given by the standard recursive formula Given a partial moving frame ρ (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 ρ (n) to obtain the recurrence relations for the partially normalized invariants where are the partially normalized horizontal one-forms and 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 relations for the phantom invariants, i.e. the lifted invariants that are set equal to a constant, by virtue of the moving frame construction. We refer the reader to [1] 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 [1]. 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 [1] 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.

Recursive Moving Frames
For a detailed exposition of the recursive moving frame implementation, we refer the reader to the original work [13]. 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 implement 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 action, the key idea of the recursive moving frame implementation is to use the recurrence formulas and the expressions for the Maurer-Cartan forms (

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 ρ (n) has been computed using the cross-section By assumption, coordinate expressions for φ α;J ν ( Z (n) ) are known, since the prolonged action up to order n has been computed, and the partially normalized Maurer-Cartan forms µ ν 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 ω i , we are able to obtain expressions for the order n + 1 partially normalized invariants U α J,j .

The Algebra of Differential Invariants
Assume a moving frame ρ (n) 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 ω i are the invariant total derivative operators be the structure equations among the invariant horizontal forms. These equations can be obtained symbolically by extending the recurrence relations (2.3) to differential forms as done in [4]. Another approach is to pull-back the Maurer-Cartan structure equations dµ = −µ ∧ µ by the (partial) moving frame ρ (n) , from which (2.7) can be deduced. Given (2.7), the commutation relations among the invariant total derivative operators are Fix j, k in (2.8) and apply the commutation relation to p invariants I 1 , . . . , which allows one to express the commutator invariants C jk in terms of I = (I 1 , . . . , I p ) and its invariant derivatives. This is what we refer to as the commutator trick. Notice that given a single invariant I 1 , we could have set I i := D ℓ i k i I 1 , with 1 ≤ k i ≤ p and ℓ i ≥ 0, in order to write the commutator invariants as functions of a single invariant and its invariant derivatives.
We now recall important results about the algebra of differential invariants that can be found in [1,11].
Proposition 2.5. The normalized invariants Z (n) provide a complete set of differential invariants of order ≤ n.
By the replacement principle, [1,8], if I(z (n) ) is a differential invariant, then it can be written in terms of the normalized invariants as I = I( Z (n) ), which is obtained by replacing the jet coordinates z (n) by their corresponding normalized invariants Z (n) . Definition 2.6. A set of invariants I gen = {I 1 , . . . , I ℓ } is said to generate the algebra of differential invariants if any differential invariant can be expressed in terms of I gen and its invariant derivatives (2.6).
From Proposition 2.5 it follows that if one can show that the normalized invariants Z (∞) can be written in terms of a set of invariants I gen and its invariant derivatives, then I gen is a generating set for the algebra of differential invariants.
Theorem 2.7. Given a moving frame ρ (n) , the normalized invariants I gen = { Z (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 I gen = { Z (n+1) } can be expressed in terms of a single invariant I and its invariant derivatives D 1 , . . . , D p , 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.

Affine Action and Low-Order Normalizations
In the following, we consider surfaces S ⊂ R 3 , which we assume are locally given a graphs of functions: We are interested in the action of the affine group A(3, R) = GL(3, R) ⋉ R 3 on these surfaces given by A basis for the algebra of infinitesimal generators is provided by denote a basis of Maurer-Cartan forms with structure equations Then the order zero recurrence relations for the lifted invariants are dX = ω x + Xµ xx + Y µ xy + Uµ xu + µ x , dY = ω y + Xµ yx + Y µ yy + Uµ yu + µ y , dU = U j ω j + Xµ ux + Y µ uy + Uµ uu + µ u , where there is no summation over k and ℓ, and δ ij denotes the Kronecker delta function.
Since the action is transitive on J (1) , we can set In other words, we can choose the cross-section K (1) = {x = y = u = u x = u y = 0} ⊂ J (1) . The recurrence relations for these phantom invariants are 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 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 Consider the partially normalized lifted Hessian determinant we conclude that H is a relative invariant. To obtain an expression for H, we introduce the determinant • 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.

Elliptic and Hyperbolic Points
In this section we work under the assumption that with ǫ = 1 corresponding to the elliptic case and ǫ = −1 to hyperbolic points. From the recurrence relations (3.3), we conclude that it is possible to set Remark 4.1. The normalization equations (4.1) are quadratic in the group parameters. In the process of constructing a moving frame there is a choice of sign that needs to be made. As is customary, [14], in the following we omit such ambiguities.
After the normalizations (4.1) have been performed, the recurrence relations for the order 3 partially normalized invariants are Consistent with normalizations performed for elliptic and hyperbolic surfaces in equi-affine geometry, [12], we set and solve for U XY 2 and U X 2 Y . We are then left with U X 3 and U Y 3 , whose recurrence relations are The extent to which one can solve for the partially normalized Maurer-Cartan forms µ yx and µ uu depends on the determinant We note that P ǫ is a relative invariant as dP ǫ = −P ǫ µ uu .
In fact, P ǫ = P a 33 , where P is the Pick invariant P = 1 16|u xx u yy − u 2 xy | 3 6u xx u xy u yy u xxx u yyy − 6u xx u 2 yy u xxx u xyy − 18u xx u xy u yy u xxy u xyy + 12u xx u 2 xy u xxy u yyy − 6u 2 xx u yy u xxy u yyy + 9u xx u 2 yy u 2 xxy − 6u 2 xx u xy u xyy u yyy + 9u 2 xx u yy u 2 xyy + u 3 xx u 2 yyy − 6u xy u 2 yy u xxx u xxy + 12u 2 xy u yy u xxx u xyy − 8u 3 xy u xxx u yyy + u 3 yy u 2 xxx .
We now need to distinguish the cases where P ǫ ≡ 0 is identically zero and where P ǫ = 0 does not vanish. In the elliptic case, we note that when P 1 ≡ 0, then U X 3 ≡ U Y 3 ≡ 0. On the other hand, in the hyperbolic case, when P −1 ≡ 0 we have that U Y 3 ≡ ±U X 3 . Thus, for hyperbolic points there are two cases to consider, either U X 3 ≡ 0 or U X 3 = 0. We combine the different cases as follows.
Case EH.1: P ǫ = 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 the three cases can be restated as

Case EH.1
When P ǫ = 0, it is possible to set According to Theorem 2.7, the order 4 differential invariants form a complete set of generating invariants. We now show in fact that the algebra of differential invariants A is generically generated by the single invariant I 1 = U Y 4 . First, the structure equations for the invariant coframe ω x , ω y are Therefore, the Lie bracket of the invariant total derivative operators is Using the commutator trick (2.9), we can generically solve for I 2 = U XY 3 and I 3 = U X 4 −2ǫU X 2 Y 2 in terms of I 1 and its invariant derivatives. Next, consider the syzygy This suggests the introduction of the fourth order invariant Also, from (4.2) is follows that I 4 can be expressed in terms of I 1 , I 2 , I 3 and their invariant derivatives. Since I 2 and I 3 and be expressed in terms of I 1 and its invariant derivatives, the same holds true for I 4 .
Now, considering the fifth order invariants D i I j , we find, using Mathematica, the syzygy This is a quintic equation in U X 2 Y 2 , which can locally be solved in terms of I 1 , I 2 , I 3 , and I 4 and their invariant derivatives. This shows the following results. Theorem 4.3. If the Pick invariant P = 0 does not vanish, then the algebra of differential invariants is generically generated by the fourth order invariant I 1 = U Y 4 .
Using the method of recursive moving frames, a coordinate expression for the generating invariant is where are invariant total derivative operators and with K a solution to the sextic equation

Case EH.2
We are now assuming that U X 3 ≡ U Y 3 ≡ 0. Their recurrence relations imply that Thus, there is only one fourth order partially normalized invariant. We continue the analysis using the invariant .
Since its recurrence relation is we now have to consider the cases When U X 2 Y 2 = 0, we can normalize U X 2 Y 2 = 1.
Using the recurrence relations, one finds that all higher order invariants are constant. Thus, there are no further normalizations possible. In particular, the Maurer-Cartan form µ = µ yx cannot be normalized. The structure equations for the coframe {ω x , ω y , µ} are

Case EH.2.2:
When U X 2 Y 2 ≡ 0, all higher order partially normalized invariants vanish. In this case µ yx and µ uu cannot be normalized and the structure equations of the coframe {ω x , ω y , µ yx , µ uu } are

Case H.3
In this section we assume that we are at a hyperbolic point where ǫ = −1. Also, we are working under the consideration that U Y 3 ≡ǫU X 3 = 0, whereǫ = ±1. Thus, it is possible to normalize U X 3 = 1. At order 4, the recurrence relation for U Y 3 −ǫU X 3 ≡ 0, yields the equalities Thus, U X 4 , U X 3 Y , and U X 2 Y 2 are functionally independent partially normalized invariants. Introducing  for k = 1, 2, 3. We now need to consider the cases Case H.3.1: Before considering each cases, we note that coordinate expressions for the invariants A i can be found using the method of recursive moving frame. We have that where Remark 4.4. The functions E ij k in (4.3) originate from the implementation of the recursive moving frame method. Namely, these functions are the horizontal components of the partially normalized Maurer-Cartan forms µ yx , µ xu , µ uu so that

Case H.3.1
In this case there is A k , with k ∈ {1, 2, 3}, such that A k = 0. For the sake of the exposition, assume A 3 = 0. The other possibilities are dealt in a similar fashion. When A 3 = 0, one can normalize A 3 = 1. Then 2 2 ]ω y , and we have the structure equations , it follows that I can be expressed in terms of A 2 and its invariant derivatives. From the syzygy , it follows that A 1 can generically be expressed in terms of A 2 and its invariant derivatives.
Theorem 4.5. The algebra of differential invariants is generically generated by the single invariant A 2 .
Remark 4.6. The coordinate expression for A 2 is obtain by solving the normalization equation A 3 = 1 for the group parameter a 33 and substituting the result into A 2 .

Case H.3.2
When A 1 ≡ A 2 ≡ A 3 ≡ 0, there is no further group parameter normalizations possible. Then, the structure equations of the coframe {ω x , ω y , µ uu } are

Parabolic Points
At a parabolic point H ≡ 0. Therefore, There are now two cases to consider. Namely, Case P.2: U X 2 ≡ 0 5.1 Case P.1 When U X 2 = 0, we can solve for U Y 2 in (5.1) to obtain Therefore, U X 2 and U XY are functionally independent partially normalized invariants. From the recurrence relations (3.3), we conclude that it is possible to set When U Y Y vanishes, the recurrence relations imply that Considering the recurrence relations for the third order partially normalized invariant U X 3 and U X 2 Y , we find From the first equation we conclude that it is possible to normalize U X 3 = 0. As for the second recurrence relation, we have the following cases to consider Introducing the ratio R = u xy u xx , a coordinate expression for U X 2 Y is given by where we recall that Y x and Y y are introduced in (3.4).

Case P.1.1
When U X 2 Y = 0, we can normalize U X 2 Y = 1. Then the recurrence relations for the non-constant fourth order partially normalized lifted invariants, i.e. U X 4 and U X 3 Y , are From the first equation, we see that it is possible to normalize U X 4 = 0. Also, the recurrence relations of the constraints (5.3) imply the following constraints on the order 5 partially normalized invariants In light of the second equation in (5.4), we now have to consider the following cases and so we can normalize U X 4 Y = 0. At this stage, the recurrence relation for the only remaining order 5 normalized invariant is We note that U X 6 and U X 5 Y are the only functionally independent order 6 invariants since From (5.5), we conclude that U X 5 Y and U X 6 can be expressed in terms of U X 5 and its invariant derivatives. It follows from Theorem 2.7 that U X 5 generates the algebra of differential invariants. Introducing the ratios we have that Finally, the structure equations of the invariant coframe {ω x , ω y } are When U X 3 Y ≡ 0, we have that and their recurrence relations further imply Thus, the recurrence relation for the only remaining order 5 partially normalized invariant, namely Thus, we have the following cases to consider In this case we normalize U X 5 = 1. Then the recurrence relation for U X 6 is and it is therefore also possible to set U X 6 = 0. At this stage all invariants of order 6 or less are constant and the only non-constant invariant of order 7 is . Since the only non-phantom invariant of order 8 is U X 8 , it follows from the recurrence relation for U X 7 that U X 7 generates the algebra of differential invariants. Finally, the structure equations are dω x = 0, dω y = 5 3 ω y ∧ ω x .

Case P.1.2
When U X 2 Y ≡ 0, we have U X 2 Y ≡ U XY 2 ≡ U Y 3 ≡ 0, which in turn implies that We also recall that U X 3 was normalized to zero. Thus, U X 4 = S a 33 u xx is the lowest order non-zero invariant left, and dU X 4 = −U X 4 µ uu mod (ω x , ω y ), which leads us to consider the following cases

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 branches of the equivalence problem under the affine group. We also provide the dimension of the self-symmetry group and recall the branches whose differential invariant algebra is generated by a single invariant. Throughout, ǫ,ǫ = ±1, with ǫ = 1 for elliptic points and ǫ = −1 for hyperbolic points.
Case EH.1: u(x, y) = 1 2 x 2 + ǫ 1 2 y 2 + 1 6 where c is the infinite vector of coefficients c i(4+j) , i, j ≥ 0 and F ij are certain universal, determinable, functions thereof. These surfaces have self-symmetry groups of dimension 0,1 of 2, depending on the particularities of c. Also, the algebra of differential invariants is generated by a single order 4 invariant.
where c is the infinite vector of c (2+i)(2+j) , i, j ≥ 0, c 22 = 0, and F ij are certain universal, determinable, functions thereof. These surfaces have self-symmetry groups of dimension 0,1 of 2, depending on the particularities of c. Furthermore, the algebra of differential invariants is generated by a single fourth order invariant.