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Equiaffine structure on frontals

In this paper, we generalize the idea of equiaffine structure to the case of frontals, in order to define the Blaschke vector field of a frontal. We also investigate the conditions that a frontal needs to satisfy in order to have a Blaschke vector field and provide some examples. Finally, taking the theory developed here into account we present a fundamental theorem, which is a version of the fundamental theorem of affine differential geometry.


Introduction
The systematic treatment of curves and surfaces in unimodular affine space is a main subject in affine differential geometry. This study has began around 1916 and mathematicians like Blaschke, Berwald, Pick and Radon were pioneers in the development of this area (see [18] for historical notes). In more recent decades, some papers are dedicated to the study of the geometry of affine immersions and structures related to affine differential geometry, as equiaffine line congruences, for instance (see [1], [5], [6], [7], [12], [21], [22], [23]).
The main goal in classical affine differential geometry is the study of properties of surfaces in 3-dimensional affine spaces, which are invariant under equiaffine transformations. In this sense, given a regular surface M and ξ an equiaffine transversal vector field on M , we obtain an equiaffine structure on M induced by ξ. By equiaffine structure, we mean a torsion free affine connection, a parallel volume element on M and a shape operator associated to ξ. Taking this into account, an important result is the fundamental theorem of affine differential geometry, which asserts that given some integrability conditions there are a surface M and an equiaffine transversal vector field ξ with a given equiaffine structure (see section 4.9 in [28] or chapter 2, section 8 in [19]).
When studying a regular non-parabolic surface M from the affine viewpoint, an important object is the associated Blaschke (or affine normal) vector field, which is an invariant under equiaffine transformations. A surface M equipped with the equiaffine structure determined by its Blaschke vector field is called a Blaschke surface. With the structure given by this vector field we define, for instance, proper and improper affine spheres, which are special types of Blaschke surfaces that have been studied in many papers (see [5], [3], [4], [17], [24], [27]).
The study of surfaces with singularities from the affine differential geometry viewpoint has not been much explored, mainly due the difficulties which arise at singular points. As we want to explore this viewpoint, in this paper we work with a special class of singular surfaces called frontals. If we take a surface S and we think of light as particles which propagate at unit speed in the direction of the normals of S, then at a given time t, this particles provide a new surface S . We call S the wave front of S. The notion of frontals arises as a generalization of wave fronts, when considering the case of hypersurfaces. In recent years, many papers are dedicated to the study of these singular surfaces (see [10], [11], [13], [14], [15], [16], [25], [26]).
Our goal is to extend the study of properties invariant under equiaffine transformations to the case of frontals, defining equiaffine structure on frontals, equiaffine transversal vector fields and the associated conormal vector field. With this, we seek to obtain a version for frontals of the fundamental theorem of affine differential geometry for regular surfaces in a way that its proof relies on assuming the integrability conditions in the regular case. We also want to understand when it is possible to define a vector field along a frontal that, at regular points, plays the same role as the classical Blaschke vector field.
In order to accomplish our goals, in section 3 we review some content about frontals and investigate some classes which play an important role in the next sections. In section 4, we generalize the idea of equiaffine structure on frontals. Since we have the notion of equiaffine transversal vector field to a frontal, in section 5 we define the Blaschke vector field of a frontal as a smooth extension of the Blaschke vector field defined on its regular part and we characterize frontals which have Blaschke vector field in theorem 5.1. Finally, in section 6 we provide in theorem 6.1 a fundamental theorem for the theory developed in the previous sections.

Fixing notations, definitions and some basic results
We denote by U an open subset of R 2 , where u = (u 1 , u 2 ) ∈ U and for a given smooth map f : U → R n the map Df : U → M n×2 (R) is the differential of f .

2.1.
Equiaffine structure for non-parabolic regular surfaces. Let R 3 be a three-dimensional affine space with volume element given by ω(e 1 , e 2 , e 3 ) = det(e 1 , e 2 , e 3 ), where {e 1 , e 2 , e 3 } is the standard basis of R 3 . Let D be the standard flat connection in R 3 and x : U → R 3 a regular surface with x(U ) = M and ξ : U → R 3 \ {0} a vector field which is transversal to M . Then, where x(u) = p. If X and Y are vector fields on M , then we have the decomposition where ∇ is the induced affine connection and c is the affine fundamental form induced by ξ. We say that M is non-degenerate if c is non-degenerate which is equivalent to say that M is a non-parabolic surface (see chapter 3 in [20]). Furthermore, we have where S is the shape operator and τ is the transversal connection form. We say that ξ is an equiaffine transversal vector field if τ = 0. We can define a volume element θ on M as follows where X and Y are tangent to M . The next proposition relates the induced volume element θ and the definition of equiaffine transversal vector field.
We say that M has a parallel volume element if there is a volume element θ on M such that ∇θ = 0, where for X, X 1 , X 2 vector fields on M . Then, it follows from proposition 2.1 that a vector field ξ, transversal to a non-parabolic surface, is equiaffine if and only if the induced volume element is parallel.
Given a non-parabolic surface x : U → R 3 the affine fundamental form c is non-degenerate, then it can be treated as a non-degenerate metric (not necessarily positive-definite) on The volume element θ induced by ξ coincides with the volume element ω c of the non-degenerate metric c.
Remark 2.1. The Blaschke vector field of a regular non-parabolic surface M is unique up to sign and it is given by ξ(u) = |K| 1/4 n + W , where K is the Gaussian curvature of x and W is a tangent vector field satisfying where II denotes the second fundamental. From (3) we can express W using the coefficients of the second fundamental form and the derivatives of |K| 1/4 as W = ax u 1 + bx u 2 , where For details, see page 45 in [20].
From now on, we refer to the Blaschke vector field given in definition 2.1 as the usual Blaschke vector field.

Frontals
A smooth map x : U → R 3 is said to be a frontal if, for all q ∈ U there is a vector field n : U q → R 3 where q ∈ U q is an open subset of U , such that n = 1 and x u i (u), n(u) = 0, for all u ∈ U q , i = 1, 2. This vector field is said to be a unit normal vector field along x. We say that a frontal x is a wave front if the map (x, n) : U → R 3 × S 2 is an immersion for all q ∈ U . Here, we consider mainly proper frontals, that is, frontals x for which the singular set Σ(x) = {q ∈ U : x is not immersive in q} has empty interior. This is equivalent to say that U \ Σ(x) is an open dense set in U .
Definition 3.1. We call moving basis a smooth map Ω : U → M 3×2 (R) in which the columns w 1 , w 2 : U → R 3 of the matrix Ω = w 1 w 2 are linearly independent vector fields.
Definition 3.2. We call a tangent moving basis (tmb) of x a moving basis Ω = (w 1 , w 2 ), such that x u 1 , x u 2 ∈ w 1 , w 2 R , where , R denotes the linear span R-vector space.
Given a tmb Ω = w 1 w 2 we denote by T Ω (q) the plane generated by w 1 (q) and w 2 (q), for all q ∈ U . Note that given two tangent moving basis Ω and Ω of a proper frontal, we get T Ω = T Ω . The next proposition provides a characterization of frontals in terms of tangent moving basis. Then, x is a frontal if and only if, for all q ∈ U , there are smooth maps Ω : U q → M 3×2 (R) and Λ : U q → M 2×2 (R) with rank(Ω) = 2 and U q ⊂ U a neighborhood of q, such that Dx(q) = ΩΛ T Ω , for allq ∈ U q . Since a tangent moving basis exist locally and we want to describe local properties, from now on we suppose that for a given frontal we have a global tangent moving basis. Then, if a frontal x satisfies Dx = ΩΛ T , where Ω is a tangent moving basis, we have that Σ(x) = λ −1 Ω (0), where λ Ω := det Λ.
Let x : U → R 3 be a frontal, Ω = w 1 w 2 a tmb and denote by n = w 1 × w 2 w 1 × w 2 the unit normal vector field induced by Ω. We set the matrices Given a frontal x : U → R 3 and a tmb Ω = w 1 w 2 it follows that w 1 , n = w 2 , n = 0. By taking this, we can rewrite Definition 3.3. Let x : U → R 3 be a frontal and Ω a tangent moving basis of x. We define the Ω-relative curvature K Ω := det(µ Ω ).
Given a frontal x : U → R 3 with a global unit normal vector field n : U → R 3 , we can also consider the matrices If we decompose Dx = ΩΛ T , then I = ΛI Ω Λ T and II = ΛII Ω . Also, the classical normal curvature at a regular point q ∈ U is given by where ϑ ∈ R 2 \ {0} are the coordinates of a vector in the tangent plane in the basis x u 1 x u 2 .
3.1. Frontals with extendable Gaussian curvature. Now, taking into account the results in [15], we investigate some classes of frontals for which the Gaussian curvature admits a smooth extension. These classes play an important role in section 5, where we define the Blaschke vector field of a frontal.
3.1.1. Frontals with extendable normal curvature. Let x : U → R 3 be a proper frontal with extendable normal curvature. It follows from corollary 3.1 in [15] that if x : (U, 0) → (R 3 , 0) is a frontal with extendable normal curvature and 0 a singularity of rank 1, then its Gaussian curvature has a smooth extension. Furthermore, this extension is non-vanishing if and only if x ∼ n, where n is the unit normal vector field of x and ∼ indicates that there are Ω 1 and Ω 2 tmb of x and n, respectively and a smooth matrix The next theorem characterizes proper frontals of rank 1 with extendable normal curvature.
where b, h, l, r are smooth function on neighborhoods of the origin in each case.

Wave fronts with extendable Gaussian curvature.
If we look at a germ of wave front x : (U, 0) → (R 3 , 0), such that 0 ∈ Σ(x) and rank Dx(0) = 1, then it follows from remark 4.1 in [15] that up to an isometry, x is R-equivalent to By R-equivalence we mean that there is a germ of diffeomorphism h : (Ũ , 0) → (U, 0) such that y = x • h. Note that Dy has decomposition Dy = ΩΛ T , where for h, g 1 and b smooth functions such that g 1 = h u 1 and −b = h u 2 . From this, we obtain that From corollary 4.3 in [15], it follows that if h in (15) satisfies the equation is a smooth function, then y is a wave front of rank 1 with extendable Gaussian curvature. Furthermore, using this information in (17) and taking into account that in U \ Σ(x) the Gaussian curvature is K = K Ω λ Ω , we get, by the density of U \ Σ(x), that its extension is given by Thus, if c(u 1 , u 2 ) is a non-vanishing function, we obtain a wave front of rank 1 for which the Gaussian curvature has a non-vanishing extension. It follows from proposition 3.4 in [15] that a wave front does not admit a smooth extension for its normal curvature, hence this class has no intersection with the class 3.1.1.
Remark 3.1. Note that the second order linear PDE is an important step in order to obtain frontals with extendable non-vanishing Gaussian curvature. If c(u 1 , u 2 ) = 1, then the equation (19) is the two dimensional Laplace equation (an elliptic equation), which was discovered by Euler in 1752. This PDE has been useful in many areas, like gravitational potential, propagation of heat, electricity and magnetism (for more information see chapter 7 in [9]). On the other hand, if we take c(u 1 , u 2 ) = −a 2 , where a = 0 is a constant, then (19) is the one-dimensional wave equation (a hyperbolic equation). The wave equation governs the dynamics of some physical systems, for instance, the guitar string, the longitudinal vibrations of an elastic bar, propagation of acoustic, fluid, and electromagnetic waves (see chapter 5 in [9]). Note in (17) that the sign of K is determined by the sign of c(u 1 , u 2 ), but this sign also identifies if the PDE (19) is hyperbolic or elliptic.

Equiaffine structure on frontals
In this section we seek to generalize the idea of equiaffine structure to the case of frontals. We define equiaffine transversal vector fields to a frontal similarly as defined in [19] when considering regular surfaces, but as we are dealing with frontals, we need to take into account tangent moving basis and the limiting tangent planes.
4.1. The case of the unit normal vector field. Let x : U → R 3 be a frontal, Ω : U → M 3×2 (R) a tmb of x, where Ω = w 1 w 2 and n : U → R 3 the unit normal vector field along x. For each q ∈ U we decompose Using this decomposition we get where the symbols T k ij , i, j, k = 1, 2 are those in (8) and (9). Note that p ij = (w i ) u j , n , thus the matrix p ij coincide with the matrix (10).
, then the matrix of p Ω relative to the basis Ω is II Ω and p Ω is nondegenerate if and only if II Ω is non-singular, which is equivalent to say that the Ω-relative curvature K Ω is non-zero (see definition 3.3).
Definition 4.1. A proper frontal is said to be a non-parabolic frontal if for some tmb Ω the relative curvature K Ω never vanishes. Remark 4.2. It follows from corollary 3.23 in [16] that a frontal x : U → R 3 is a wavefront if and only if, (K Ω , H Ω ) = 0 on Σ(x), for whatever tangent moving base Ω. Therefore, every non-parabolic frontal is a wavefront.
Proposition 4.1. The notion of non-parabolicity is independent of tmb.
Proof. Via proposition 3.18 in [16] the zeros of K Ω are independent of tmb.
The next proposition provides a representation formula for non-parabolic frontals.
be a germ of non-parabolic frontal, Ω a tmb of x and 0 ∈ Σ(x). Then, up to an isometry x is R-equivalent to where a, b are smooth functions such that a u 2 = b u 1 .

4.2.
The case of a transversal vector field. Let ξ : U → R 3 be a vector field which is transversal to the frontal x : In the same way that occurs with the unit normal vector field, we write and we obtain a bilinear form h Ω (q) : T Ω × T Ω → R, such that h Ω (q)(w i , w j ) = h ij (q). We call h Ω the relative affine fundamental form of x induced by ξ. In a similar way, we write Then for each q ∈ U we have S Ω (q) : We call S Ω the relative shape operator of ξ and τ Ω the relative transversal connection form.
Definition 4.2. The vector ξ defines an equiaffine structure on x (or ξ is equiaffine) when the derivatives of ξ are in T Ω (q), for all q ∈ U , i.e. when τ Ω ≡ 0.
For a frontal x : U → R 3 and an equiaffine transversal vector field ξ : U → R 3 we say that the symbols D k ij and h ij , given in (24), define an equiaffine structure on x.
whereω is the canonical volume element in R 3 , that isω(e 1 , e 2 , e 3 ) = det(e 1 , e 2 , e 3 ), for {e 1 , e 2 , e 3 } the standard basis of R 3 . The volume element θ is called the induced volume element. Proof. Let ξ be a transversal vector field, thus we can write From this, p ij is non-singular if and only if h ij is non-singular. As x is non-parabolic, the result follows from remark 4.1.
Let us suppose that Note that (a) was proved in proposition (4.3). In order to prove (b) let us write B Ω = b ij the relative shape operator of n. We know that On the other hand, where Z u i is the tangent component of Z u i . By comparing the normal components of (26) and (27), it follows that τ Ω ( Let us take x : U → R a frontal and Ω = w 1 w 2 a tmb. If ξ : U → R 3 , given by ξ = φn + aw 1 + bw 2 , is an equiaffine transversal vector field, then τ Ω ≡ 0. Via Note in remark 2.1 that we could write the vector field W considering a tmb Ω = w 1 w 2 instead of the usual tmb Dx = x u 1 x u 2 . In this case, we would obtain an expression like (28) instead of (4).

The Blaschke and the conormal vector fields of a frontal
The Blaschke vector field and the conormal vector field associated to an equiaffine transversal vector field play an important role when studying regular surfaces from the affine differential geometry viewpoint. With the Blaschke structure, for instance, we define proper and improper affine spheres (see chapter 2 in [19]) and the Blaschke line congruences (see section 6 in [12]). On the other hand, the conormal vector field makes calculations with the affine support function easier, for instance (see section 1 in [2]). Taking into account the importance of these two objects, in this section we define the Blaschke vector field of a frontal and the conormal vector field associated to an equiaffine vector field transversal to a frontal. 5.1. The Blaschke vector field of a frontal. Definition 5.1. Let x : U → R 3 be a proper frontal such that its Gaussian curvature never vanishes in U \ Σ(x). We say that a transversal vector field ξ is the Blaschke vector field of x if it is a smooth extension of the usual Blaschke vector field defined in U \ Σ(x).
It follows from the density of U \ Σ(x) and from the fact that the Blaschke vector field is unique up to sign (see [20]) that the above extension is unique. Now, looking at the special affine group (or equiaffine group) SA(3, R) = {Φ : x → Ax + b : A ∈ M 3 (R), det A = 1 and b is a constant vector} we seek to show an invariance property for the Blaschke vector field defined here, in the following sense. Given a frontal x : U → R 3 and Φ ∈ SA(3, R), the Blaschke vector field of Proposition 5.1. Let x : U → R 3 be a proper frontal for which the Gaussian curvature has a non-vanishing smooth extension. If there exists, the Blaschke vector field of x is an equiaffine invariant.
Proof. Let Φ(x) = Ax + b be an equiaffine transformation, then y = Φ • x has the same singular set of x. Furthermore, it is known that in U \ Σ(x) the usual Blaschke vector field of y is given by ξ = Φ * ξ, where ξ is the Blaschke vector field of x. Let us keep the same notation for the extension of ξ, then Φ * ξ is an extension of ξ to U , so it is the Blaschke vector field of y. Remark 5.1. Note that is not always possible to obtain a smooth extension of the Blaschke vector field. For instance, if x : U → R 3 is a non-parabolic frontal, i.e., for all tmb Ω, the Ω-relative curvature K Ω = 0, then the Gaussian curvature is not extendable (see proposition 4.1 in [14]). It is known that the usual Blaschke vector field is given in U \Σ(x) by |K| 1/4 n+W , where W is a tangent vector field (see remark 2.1), thus ξ, n = |K| 1/4 . Since it is not possible to extend K, it follows that is not possible to extend this vector field. From this, it follows that a necessary condition to obtain the Blaschke vector field of a frontal is that its Gaussian curvature is extendable.
Next, we characterize frontals for which it is possible to define the Blaschke vector field.
Theorem 5.1. A frontal x : U → R 3 admits a Blaschke vector field if and only if its Gaussian curvature K has a non-vanishing extension to U and there are a, b ∈ C ∞ (U, R) such that where Ω = w 1 w 2 is a tmb and φ = |K| 1/4 .
Proof. Let ξ = φn + aw 1 + bw 2 be the Blaschke vector field of x. Then, φ, a, b ∈ C ∞ (U, R) and φ = ξ, n = |K| 1/4 in U \ Σ(x) (see remark 2.1). It follows from the density of U \ Σ(x) and from the smoothness of φ that φ(q) = lim u→q |K| 1/4 . From the fact that ξ is transversal to x, it follows that φ = 0 in U , consequently K admits a non-vanishing extension to U . As ξ is equiaffine in U \ Σ(x), it follows from remark 4.3 that Since a, b ∈ C ∞ (U, R) and U \ Σ(x) is dense, we obtain for a point q ∈ Σ(x) that Reciprocally, considering K the non-vanishing extension of the Gaussian curvature of x, define φ = |K| 1/4 and take a, b ∈ C ∞ (U, R) satisfying (30), then it follows from definition 5.1 and from remarks 2.1 and 4.3 that ξ is the Blaschke vector field of x.

Remark 5.2.
(a) Let x : U → R 3 be a frontal in the class 3.1.1, such that its Gaussian curvature K admits a non-vanishing extension, so in order to have a Blaschke vector field we just need to verify the condition (30) in theorem 5.1. One can verify that K u 2 = db u 2 , for a smooth function d is a sufficient condition for this to happen. For instance, with any of the choices below • b = u 2 2 , r = 0, l = 1 and h = h(u 1 , u 2 ) any smooth function (see example 5.1), • b = 2 5 u 5 2 + u 2 2 , r = 0, l = 1 and h = h(u 1 , u 2 ) any smooth function (see example 5.2), we obtain x for which K admits a non-vanishing extension and condition (30) is verified, hence we have a Blaschke vector field.
(b) If x : U → R 3 is a wave front given by the class 3.1.2, such that the Blaschke vector field exists in the regular part U \ Σ(x), then it is given by ξ = (ξ 1 , ξ 2 , ξ 3 ), where Thus, if we take c constant, we get ξ = (0, 0, ρ), for some ρ ∈ R. Note in (33) that is not always possible to extend ξ, since h u 1 u 1 (q) = 0 for all q ∈ Σ(x). However, if we take for instance, c a smooth function such that c u 1 = dh u 1 u 1 , for a smooth function d, then ξ admits an extension to the entire U . Note that if d = 0, we get c = c(u 2 ), which is satisfied for the case c constant, for instance (see example 5.3).

Figure 5.
Wave front of rank 1 with extendable non-vanishing Gaussian curvature.
Remark 5.4. It follows from the second condition above that the conormal vector field is always a multiple of the unit normal vector field n of x.
The next proposition shows that an important property of the conormal vector field is still valid when we are working with the case of non-parabolic frontals.
Proposition 5.2. Given a frontal x : U → R 3 , a tmb Ω and an equiaffine transversal vector field ξ, we have that Furthermore, if x is non-parabolic then the conormal vector field is an immersion.
Proof. In order to simplify notation, we drop the subscript in the notation for the induced affine fundamental form and indicate only by h. We know that ν, ξ = 1 so, differentiating and using the fact that ξ is equiaffine, we get ν u i , ξ = 0. From where v u i indicates the tangent component of v u i , thus considering the properties of the conormal vector field, we obtain Now, let us suppose that x is non-parabolic and that ν is not an immersion, hence there is (a, b) ∈ R 2 \ 0 such that aν u 1 + bν u 2 = 0. Thus, for all v ∈ T x,q , we have that By using (35) and the above expression, we get but this contradicts the fact that h is non-degenerate.

A fundamental theorem
In this section we provide, in theorem 6.1, a fundamental theorem for the theory developed in section 2. This theorem is a version for frontals of the fundamental theorem of affine differential geometry (see section 4.9 in [28] for the classical result for regular surfaces). Thus, taking U ⊂ R 2 an open subset and assuming the integrability conditions for the regular case are valid in an open dense subset of U , we obtain for each q ∈ U a neighborhood V ⊂ U of q, a frontal x : V → R 3 and an equiaffine transversal vector field ξ : V → R 3 , in the sense of section 2. In order to do this, we use the same approach applied in [16].
6.1. The compatibility equations. Let x : U → R 3 be a proper frontal, Ω = w 1 w 2 a tmb and n = w 1 ×w 2 w 1 ×w 2 the unit normal vector field induced by Ω. By considering the decomposition Dx = ΩΛ T and λ Ω = det Λ, one can get in U \ λ −1 Ω (0) the following structural equations where e, f and g are the coefficients of the second fundamental form of x. In a similar way, if ξ : U → R 3 is an equiaffine transversal vector field to x, the following hold in U \ λ −1 The symbols c ij induce a symmetric bilinear form called the affine fundamental c relative to ξ, while the symbols Γ k ij are associated to the induced affine connection ∇ (see (1)). Proposition 6.1. If we write ξ = φn + ax u 1 + bx u 2 , where φ = 0, then in U \ Σ(x) we have: Proof. If ξ = φn + ax u 1 + bx u 2 in (37), then Hence, comparing this to (36) we have the result.
If we look at the basis w 1 , w 2 , n of R 3 , it follows from (21) that there are smooth functions T k ij defined in U , i, j, k ∈ {1, 2}, such that w 1u 1 = T 1 11 w 1 + T 2 11 w 2 + e Ω n (41a) We know that w 1 , w 2 , ξ is a basis of R 3 , then from (24), (25) and from the fact that ξ is equiaffine there are smooth functions D k ij , h ij and S i j defined in U , i, j, k ∈ {1, 2}, such that w 1u 1 = D 1 11 w 1 + D 2 11 w 2 + h 11 ξ (42a) If we write ξ = φn + ax u 1 + bx u 2 and ξ = φn +ãw 1 +bw 2 , where φ = 0, then in U \ Σ(x) Hence, taking (41), (42), (43) and using the method of the proof of proposition 6.1 one can get the following result: Proposition 6.2. If we write ξ = φn +ãw 1 +bw 2 , for φ,ã,b ∈ C ∞ (U, R) and φ = 0, then where D 1 = Remark 6.1. With notation as in proposition (6.2), it follows from II = ΛII Ω (see 12), from 38 and from (44) that in U \ Σ(x). Furthermore, just taking the decomposition Dx = ΩΛ T , we have that Proposition 6.3. Let x : U → R 3 be a proper frontal and Ω a tmb of x. Then, in U \ Σ(x), we can write Proof. It is known that the Christoffel symbols for the decomposition in the basis {x u 1 , x u 2 , n} are given by Γ 1 = Dx T u 1 Dx I −1 (see section 4.3 in [8]) thus, by taking (38), we get From now on, if A ∈ M n×n (R), we denote by A (i) the ith-row and by A (j) the jth-column of A. Proposition 6.4. Let I, I Ω , Λ, h ij , c ij : U → M 2×2 (R) be arbitrary smooth maps, such that I Ω is symmetric and i, j = 1, 2. Consider also λ Ω = det Λ and T Ω the principal ideal generated by λ Ω in the ring is an open dense set and that where φ ∈ C ∞ (U, R \ 0) and define in U \ λ −1 Ω (0), Γ 1 and Γ 2 by (39) and (40), respectively. Then, has a unique C ∞ extension to U if and only if, has a unique C ∞ extension to U if and only if, Proof.
(a) Let us suppose D 1 the C ∞ extension of Λ −1 Γ 1 Λ − Λ u 1 to U , thus . From (39), it follows that It is known that Via (52) we have an expression for I u 1 , then multiplying the above expression by the right side with 2I Ω Λ T and taking into account (45) and (47), we obtain that Note that the right side of the above equation is a skew-symmetric matrix, which Multiplying (56) by the left side with 1 0 and by the right side with 0 1 T , we Reciprocally, suppose (54). Since U \ λ −1 Ω (0) is a dense set, there is a unique ω 1 ∈ C ∞ (U, R) such that From the above expression, it follows that ΛI Ω Λ t Taking into account (52) and then the expression for I u 1 we show, using the above expression, that is dense, we have the result. Item (b) follows analogously by considering the matrix In preparation for the Fundamental theorem, let us set the matrices W = w 1 w 2 ξ ∈ GL(3), Then, the system (42) is represented by It is known that the compatibility condition for the system (61) is W T u 1 u 2 = W T u 2 u 1 , from which we obtain We have next two auxiliary lemmas, which play an important role in the proof of theorem 6.1.
Lemma 6.1. The integrability conditions for the system      Proof. It is known that the integrability condition for the system (64) is x u 1 u 2 = x u 2 u 1 . If we set the matrices so, the integrability condition is equivalent to D T 1 Λ Tĵ + Λ T u 1ĵ = D T 2 Λ Tî + Λ T u 2î . By taking each component of this expression we have Finally, note that (66) and (67) are equivalent to 0 1 (ΛD 1 + Λ u 1 ) = (ΛD 2 + Λ u 2 ) 1 0 and (68) where all the components above are C ∞ functions defined in U , λ Ω = det Λ and U \ λ −1 Ω (0) is a dense set, for Λ = λ ij . Suppose also that the compatibility equations for the system (37) are satisfied in U \ λ −1 Ω (0) and that where T Ω is the principal ideal generated by λ Ω in the ring C ∞ (U, R). Then, (a) For each q ∈ U , there exists a neighborhood V ⊂ U of q, a frontal x : V → R 3 with tmb Ω, such that Dx = ΩΛ T , and an equiaffine transversal vector field ξ : V → R 3 with associated equiaffine structure given by (72) 1 φ where D 1 and D 2 are the unique C ∞ extensions of Λ −1 Γ 1 Λ − Λ u 1 and Λ −1 Γ 2 Λ − Λ u 2 to U , respectively.
(b) If moreover, we suppose eg − f 2 = 0 in V \ λ −1 Ω (0) and that the condition ∇ω c = 0 is satisfied in V \ λ −1 Ω (0), where ∇ is the connection associated to the symbols and ω c is the volume element induced by the affine fundamental form c, then there is a volume element ω in R 3 such that ξ is the Blaschke vector field of the frontal x. (c) Let U be connected. Suppose that x : U → R 3 is another proper frontal, ξ an equiaffine transversal vector field and Ω a tmb satisfying the same conditions that were obtained in (a). Then, x e x are affinely equivalent. Proof.
(a) It follows from (71) and from proposition (6.4) that the maps defined in U \ λ −1 Ω (0), admit unique C ∞ extensions to U , D 1 e D 2 , respectively. Thus, in U \ λ −1 Ω (0), we have By using D 1 and D 2 , we can build the matrices D 1 and D 2 as the matrices (60). Then, using (69) and (70) we have that Let us consider the following system of PDE where v 1 , v 2 , v 3 are linearly independent vectors in R 3 and q ∈ U is a fixed point. The compatibility conditions for the system (37) are expressed by Γ 1u 2 − Γ 2u 1 + Γ 1 , Γ 2 = 0 and by hypothesis they are satisfied, so it follows from lemma 6.2 that D 1u 2 − D 2u 1 + [D 1 , D 2 ] = 0, which is equivalent to the compatibility conditions for the system (77) (see 61). Thus, this system has a unique solution W : V → GL (3), where V ⊂ U is a neighborhood of q. If W = w 1 w 2 ξ , it follows that the vector field ξ is transversal to w 1 , w 2 R and that ξ u i ∈ w 1 , w 2 R , i = 1, 2. Now, let us take the following system of PDE      x u 1 = λ 11 w 1 + λ 12 w 2 x u 2 = λ 21 w 1 + λ 22 w 2 x(q) = p, for a fixed p ∈ R 3 . Note that 0 1 (ΛD 1 + Λ u 1 ) = 0 1 Γ 1 Λ = Γ 2 Λ 1 0 = (ΛD 2 + Λ u 2 ) 1 0 in λ −1 Ω (0) c , so, via density the equality holds in U . The above equality, together with the fact that Λ h 11 h 12 h 21 h 22 is symmetric, means that the system (78) has a unique solution x : V → R 3 , where V ⊂ V is a neighborhood of q (see lemma 6.1). As Dx = ΩΛ T , where Ω = w 1 w 2 it follows that x is a frontal for which Ω is a tmb, ξ is an equiaffine transversal vector field and the equiaffine structure is the desired one. (b) Moreover, from eg − f 2 = 0 in V \ λ −1 Ω (0) we obtain that c is non-degenerate in V \ λ −1 Ω (0), thus let ω c be the volume element induced by the non-degenerate metric c. Let ω 1 be the volume element in R 3 , so the volume element induced by the equiaffine vector field ξ is θ 1 (v 1 , v 2 ) = ω 1 (v 1 , v 2 , ξ), where v 1 , v 2 are tangent vectors. Since ξ is equiaffine, θ 1 is a parallel volume element in the regular part of x, i.e., ∇θ 1 = 0 (see proposition 2.1). The apolarity condition ∇ω c = 0 means that ω c = µθ 1 , where µ is a positive constant, since a parallel volume element is unique up to a positive scalar multiple. If we take in R 3 the volume element ω = µω 1 and the new induced volume element θ, we get ω c = θ, therefore, ξ is the usual Blaschke vector field of x in the regular part. Since ξ is defined in V , it follows that ξ is the Blaschke vector field of x as defined in 5.1. (c) If we write Ω = w 1 w 2 and Ω = w 1 w 2 , then for each q ∈ U there is an isomorphism L q : R 3 → R 3 , such that L q (w i ) = w i , i = 1, 2.
Therefore, L is constant and from Ω = LΩ we obtain that is, x = Lx + a, where a is a constant vector and L : R 3 → R 3 is a linear isomorphism.