EQUIAFFINE STRUCTURE ON FRONTALS IGOR CHAGAS SANTOS Instituto de Ciˆ encias Matem aticas e de Computa c ao Universidade de S ao Paulo Av.

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EQUIAFFINE STRUCTURE ON FRONTALS

IGOR CHAGAS SANTOS

Instituto de Ciˆencias Matem´aticas e de Computa¸c˜ao, Universidade de S˜ao Paulo, Av.

Trabalhador S˜ao-carlense, 400, S˜ao Carlos, SP 13566-590, Brazil.

Abstract.

In this paper, we generalize the idea of equiaﬃne structure to the case of frontals

and we deﬁne the Blaschke vector ﬁeld of a frontal. We also investigate some necessary

and suﬃcient conditions that a frontal needs to satisfy 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 for frontals of the fundamental theorem of aﬃne

diﬀerential geometry.

Keywords: Frontals; Blaschke vector ﬁeld; Equiaﬃne structure.

MSC: 53A15, 53A05, 57R45.

1. Introduction

The systematic treatment of curves and surfaces in unimodular aﬃne space is a main

subject in aﬃne diﬀerential geometry. This study began around 1916 and mathematicians

like Blaschke, Berwald, Pick and Radon were pioneers in the development of this area (see

[

] 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

line congruences, for instance (see [1], [2], [6], [7], [14], [24], [27], [28]).

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

and

an equiaﬃne transversal vector ﬁeld on

, we obtain

an equiaﬃne structure on

induced by

. By equiaﬃne structure, we mean a torsion free

aﬃne connection, a parallel volume element on

and a shape operator associated to

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

and

an equiaﬃne transversal vector ﬁeld

with a given equiaﬃne structure (see section 4.9 in [

]

or chapter 2, section 8 in [25]).

When studying a regular non-parabolic surface

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

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 [6], [4], [5], [21], [29], [32]).

The study of surfaces with singularities from the aﬃne diﬀerential geometry viewpoint has

not been much explored, mainly due to the diﬃculties which arise at singular points. As we

E-mail address:igor.chs34@gmail.com.

The author was supported by CAPES Proc. PROEX-11365975/D.

Present address: Instituto de Matem´atica e Estat´ıstica, Universidade Federal da Bahia, Rua Br.

de Jeremoabo, 40170-115 Salvador, BA, Brazil.

arXiv:2210.10847v3 [math.DG] 5 Feb 2025

EQUIAFFINE STRUCTURE ON FRONTALS 2

want to explore this viewpoint, in this paper we work with a special class of singular surfaces

called frontals. If we take a surface

and we think of light as particles which propagate at

unit speed in the direction of the normals of

, then at a given time

, this particles provide a

new surface

S′

. We call

S′

the wave front of

. 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 [

], [

[31]). Other references can be found in the survey paper [11].

Our goal is to extend the study of properties invariant under equiaﬃne transformations to

the case of frontals. We ﬁrst deﬁne the notion of equiaﬃne transversal vector ﬁeld to a frontal,

using the limiting tangent planes. With this, we provide a deﬁnition of aﬃne non-degenerate

frontal and deﬁne the Blaschke vector ﬁeld of such a frontal as the smooth extension of the

Blaschke vector ﬁeld deﬁned on its regular part. In theorem 5.1 some necessary and suﬃcient

conditions that a frontal needs to satisfy to have a Blaschke vector ﬁeld are shown.

In remark 5.2 we provide some classes of frontals that admit a Blaschke vector ﬁeld. When

studying the class of wave fronts of rank 1 with extendable Gaussian curvature it turns out

that from this class we get a subclass of frontal improper aﬃne spheres, i.e. frontals with

constant Blaschke vector ﬁeld. This class seems to be related to improper aﬃne maps, that

is a class of improper aﬃne spheres with singularities introduced in [

] for convex surfaces.

It is worth observing that improper aﬃne spheres with singularities is a topic of interest in

diﬀerential geometry, see [6], [13], [20] and [22], for instance.

Finally, taking into account the equiaﬃne theory developed here, we obtain in theorem 6.1

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. To prove this theorem the same approach used in [

] is applied, but here we are working

not only with the unit normal vector ﬁeld, but with any equiaﬃne vector ﬁeld transversal to

a frontal.

This paper is organized as follows. In section 2 some well known results from the equiaﬃne

theory for regular surfaces are shown. 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

and we characterize frontals which have Blaschke vector ﬁeld. Finally, in section 6 we provide

a fundamental theorem for the theory developed in the previous sections.

Acknowledgements. This work is part of author’s Ph.D thesis, supervised by Maria Aparecida

Soares Ruas and D´ebora Lopes, whom the author thanks for all the support and constant

motivation. The author thanks Tito Medina for his support and also for being constantly

available for helpful discussions. The author is also grateful to Ra´ul Oset and the Singularity

Group at the Universitat de Val`encia for their hospitality and useful comments on this work

during author’s stay there.

Statements and Declarations.

Availability of data. Data sharing not applicable to this article as no datasets were generated

or analyzed during the current study.

Conﬂict of interest. The author has no conﬂicts of interest to declare that are relevant to the

content of this study

EQUIAFFINE STRUCTURE ON FRONTALS 3

2. Fixing notations, definitions and some basic results

We denote by

an open subset of

, where

= (

u1, u2

)

∈U

and for a given smooth map

f:U→Rnthe map Df :U→Mn×2(R) is the diﬀerential of f.

2.1. Equiaﬃne structure for non-parabolic regular surfaces. Let

be a three-

dimensional aﬃne space with volume element given by

) =

det

), for

3∈R3

. Let

be the standard ﬂat connection in

and x:

U→R3

a regular

surface with x(

) =

and

U→R3\ {

}

a vector ﬁeld which is transversal to

. Then,

TpR3=TpM⊕ ⟨ξ(u)⟩R,

where x(

) =

, for any

u∈U

. If

and

are vector ﬁelds on

, then we have the

decomposition

DXY=∇XY+c(X, Y )ξ,(1)

where

∇

is the induced aﬃne connection and cis the aﬃne fundamental form induced by

We say that

is non-degenerate if cis non-degenerate which is equivalent to say that

a non-parabolic surface (see chapter 3 in [26]). Furthermore, we have

DXξ=−S(X) + τ(X)ξ,

where

is the shape operator and

is the transversal connection form. We say that

is an

equiaﬃne transversal vector ﬁeld if

= 0. The induced volume element

is deﬁned as follows

θ(X, Y ) := ω(X, Y, ξ),

where Xand Yare tangent to M.

Deﬁnition 2.1. Let

be an arbitrary vector ﬁeld which is transversal to

,cthe aﬃne

fundamental form and

the induced volume element. We deﬁne

detθ

cas

det

), where

cij =c(Xi, Xj) and {X1, X2}is a unimodular basis for θ, that is, θ(X1, X2) = 1.

Remark 2.1. Since the determinant of

cij

is independent of the choice of unimodular basis

{X1, X2}, detθcis well deﬁned.

The next proposition relates the induced volume element

and the deﬁnition of equiaﬃne

transversal vector ﬁeld.

Proposition 2.1. ([26], Proposition 1.4) We have

∇Xθ=τ(X)θ, for all X∈TpM.(2)

Consequently, the following two conditions are equivalent:

(a) ∇θ= 0.

(b) τ= 0.

We say that

has a parallel volume element if there is a volume element

such

that ∇θ= 0, where

∇Xθ(X1, X2) = X(ω(X1, X2)) −θ(X1,∇XX2)−θ(∇XX1, X2)

for

X, X1, X2

vector ﬁelds on

. Then, it follows from proposition 2.1 that a vector ﬁeld

transversal to a non-parabolic surface, is equiaﬃne if and only if the induced volume element

is parallel.

EQUIAFFINE STRUCTURE ON FRONTALS 4

Given a non-parabolic surface x:

U→R3

the aﬃne fundamental form cis non-degenerate,

then it can be treated as a non-degenerate metric (not necessarily positive-deﬁnite) on

x(U) = M.

Deﬁnition 2.2. Let x:

U→R3

be a non-parabolic surface. A transversal vector ﬁeld

U→R3\ {

}

is the Blaschke normal vector ﬁeld of x(

) =

if the following conditions

hold:

(a) ξis equiaﬃne.

(b)

The volume element

induced by

coincides with the volume element

ωc

of the

non-degenerate metric c.

From now on, we refer to the Blaschke vector ﬁeld given in deﬁnition 2.2 as the usual

Blaschke vector ﬁeld.

3. Frontals

A smooth map x:

U→R3

is said to be a frontal if, for all

q∈U

there is a vector ﬁeld

Uq→R3

where

q∈Uq

is an open subset of

, such that

∥

= 1 and

⟨

(

)

⟩

= 0,

for all

u∈Uq

= 1

2. This vector ﬁeld is said to be a unit normal vector ﬁeld along x. We

say that a frontal xis a wave front if the map (x

n) :

U→R3×S2

is an immersion for all

q∈U

. Here, we consider mainly proper frontals, that is, frontals xfor which the singular set

Σ(x) =

{q∈U

xis not immersive at 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 a moving basis a smooth map Ω:

U→ M3×2

(

) in which the

columns w

U→R3

of the matrix Ω=

w1w2

are linearly independent vector ﬁelds.

Deﬁnition 3.2. We call a tangent moving basis (tmb) of xa moving basis Ω=

(w1,w2)

such that xu1,xu2∈ ⟨w1,w2⟩R, where ⟨,⟩Rdenotes the linear span R-vector space.

Given a tmb Ω=

w1w2

we denote by

TΩ

(

) the plane generated by w

(

) and w

(

for all

q∈U

. Note that given two tangent moving basis Ωand

Ω

of a proper frontal, we get

TΩ

Ω

. The next proposition provides a characterization of frontals in terms of tangent

moving basis.

Proposition 3.1. ([

], Proposition 3.2) Let x:

U→R3

be a smooth map with

U⊂R2

an open set. Then, xis a frontal if and only if, for all

q∈U

, there are smooth maps

Ω:

Uq→ M3×2

(

) and Λ:

Uq→ M2×2

(

) with

rank

(Ω) = 2 and

Uq⊂U

an open

neighborhood of q, such that Dx(˜q) = ΩΛT

Ω, for all ˜q∈Uq.

Remark 3.1. It follows from 3.1 that

U→R3

is a frontal if, and only if, there is a tangent

moving basis of x.

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 xsatisﬁes

x=ΩΛ

, where Ωis a tangent moving basis, we have that Σ(x) =

λ−1

Ω

(0),

where λΩ:= det Λ.

EQUIAFFINE STRUCTURE ON FRONTALS 5

Let x:

U→R3

be a frontal, Ω=

w1w2

a tmb of xand denote by n=

w1×w2

∥w1×w2∥

the

unit normal vector ﬁeld induced by Ω. We set the matrices

IΩ:= ΩTΩ=EΩFΩ

FΩGΩ=⟨w1,w1⟩ ⟨w1,w2⟩

⟨w2,w1⟩ ⟨w2,w2⟩,

IIΩ:= −ΩTDn=eΩf1Ω

f2Ω gΩ=−⟨w1,nu1⟩ −⟨w1,nu2⟩

−⟨w2,nu1⟩ −⟨w2,nu2⟩,

µΩ:= −IIT

ΩI−1

Ω,

T1:= (ΩT

u1Ω)I−1

Ω=T1

11 T2

21 T1

21,(3)

T2:= (ΩT

u2Ω)I−1

Ω=T1

12 T2

22 T1

22.(4)

Given a frontal x:

U→R3

and a tmb Ω=

w1w2

of x, it follows that

⟨

⟩

⟨

⟩

0. By taking partial derivatives of these equalities, we rewrite

IIΩ=⟨(w1)u1,n⟩ ⟨(w1)u2,n⟩

⟨(w2)u1,n⟩ ⟨(w2)u2,n⟩.(5)

Deﬁnition 3.3. Let x:

U→R3

be a frontal and Ωa tangent moving basis of x. We deﬁne

the Ω-relative curvature KΩ:= det(µΩ).

Given a frontal x:

U→R3

with a global unit normal vector ﬁeld n:

U→R3

, we can also

consider the matrices

I:= DxTDx=E F

F G=⟨xu1,xu1⟩ ⟨xu1,xu2⟩

⟨xu1,xu1⟩ ⟨xu2,xu2⟩,

II := −DxTDn=e f

f g=−⟨xu1,nu1⟩ −⟨xu1,nu2⟩

−⟨xu2,nu1⟩ −⟨xu2,nu2⟩.(6)

If we decompose

x=ΩΛ

, then I=ΛI

Ω

and

=Λ

IIΩ

. Also, the classical normal

curvature at a regular point q∈Uis given by

kq(ϑ) := ϑTIIϑ

ϑTIϑ,

where ϑ∈R2\ {0}are the coordinates of a vector in the basis xu1xu2.

Let x:

U→R3

be a frontal, Ω:

U→M3×2

(

) a tmb of x, where Ω=

w1w2

and

n:U→R3the unit normal vector ﬁeld along x. For each q∈Uwe decompose

R3=TΩ(q)⊕ ⟨n(q)⟩R.

Using this decomposition we get

wiuj=T1

ij w1+T2

ij w2+pijn,(7)

where the symbols

i, j, k

= 1

2 are those in (3) and (4). Note that

pij

⟨

)

uj,

⟩

, thus

the matrix pijcoincide with the matrix (5).

Remark 3.2. If we deﬁne a bilinear form

pΩ

(

) :

TΩ×TΩ→R

, given by

pΩ

(

)(

wi,wj

) =

pij

⟨

iuj

(

)

⟩

, then the matrix of

pΩ

relative to the basis Ωis

IIΩ

and

pΩ

is non-degenerate

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EQUIAFFINESTRUCTUREONFRONTALSIGORCHAGASSANTOSInstitutodeCiˆenciasMatem´aticasedeComputa¸c˜ao,UniversidadedeS˜aoPaulo,Av.TrabalhadorS˜ao-carlense,400,S˜aoCarlos,SP13566-590,Brazil.Abstract.Inthispaper,wegeneralizetheideaofequiaffinestructuretothecaseoffrontalsandwedefinetheBlaschkevectorfieldofafront...

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EQUIAFFINE STRUCTURE ON FRONTALS IGOR CHAGAS SANTOS Instituto de Ciˆ encias Matem aticas e de Computa c ao Universidade de S ao Paulo Av.

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