LINE CONGRUENCES ON SINGULAR SURFACES D. LOPESyT.A. MEDINA-TEJEDAM.A.S. RUASzAND I.C. SANTOS Abstract. This paper is a rst step in order to extend Kummers the-

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LINE CONGRUENCES ON SINGULAR SURFACES

D. LOPES†,T.A. MEDINA-TEJEDA?,M.A.S. RUAS‡AND I.C. SANTOS∗

Abstract. This paper is a ﬁrst step in order to extend Kummer’s the-

ory for line congruences to the case {x, ξ}, where x:U→R3is a smooth

map and ξ:U→R3is a proper frontal. We show that if {x, ξ}is a

normal congruence, the equation of the principal surfaces is a multiple of

the equation of the developable surfaces, furthermore, the multiplicative

factor is associated to the singular set of ξ.

1. INTRODUCTION

A line congruence in the Euclidean space of dimension 3 is a 2−parameter

family of lines in R3.

The ﬁrst record about line congruences appeared in “M´emoire sur la

Th´eorie des D´eblais et des Remblais” (1776,1784) where Gaspard Monge

seeks to solve a minimizing cost problem of transporting an amount of land

from one place to another, preserving the volume (see [5] for historical notes).

After Monge, Ernst Eduard Kummer in “Allgemeine Theorie der geradlini-

gen Strahien systeme” (see [4]) was the ﬁrst to deal with the general theory

of line congruences. This theory is currently known as Kummer theory of

line congruences. In recent years, the subject achieved an important devel-

opment with contributions by [2], [6], [7], [8], [15] among others.

A line congruence is deﬁned by a pair {x, ξ} ∈ C∞(U, R3×R3\{0}) where

U⊂R2is an open set and for all (u1, u2)∈U,ξ(u1, u2) denotes the direction

of the line through the point x(u1, u2). An example of line congruences is

given by the normal lines of a surface in R3. Each smooth curve Cin x(U)

is the directrix curve of a ruled surface whose generators are the lines of

the congruence passing through points of C. These ruled surfaces are called

surfaces of congruences. Line congruences and ruled surfaces are often used

in optics, mechanics, space kinematics and robot motion planning ([9], [16]).

Izumiya, Saji and Takeuchi, in [7], represent (locally) a line congruence

as the image of the map F(x,ξ):U×I→R3where F(x,ξ)(u1, u2, t) =

x(u1, u2) + tξ(u1, u2), x:U→R3, ξ :U→R3− {0}are smooth mappings,

†Partially supported by INCTMat/CAPES Proc. 88887.510549/2020-00.

?Supported by FAPESP Proc. 2021/11253-3.

‡Partially supported by FAPESP Proc. 2019/21181-0 and CNPq Proc. 305695/2019-3.

∗Supported by CAPES Proc. PROEX-11365975/D.

arXiv:2210.14175v1 [math.DG] 25 Oct 2022

LINE CONGRUENCES ON SINGULAR SURFACES 2

U⊂R2is an open region and Iis an open interval. In this sense, we can

see that line congruences generalize the concept of parallel surfaces.

Most known results in Kummer’s theory are formulated for congruences

{x, ξ}where xis a regular surface and ξis an immersion. For instance, we

discuss in Proposition 2.2 a nice way of deﬁning lines of curvature using line

congruences: lines of curvature on a smooth surface are those curves whose

surfaces of congruence SCare developable.

Our goal in this paper is to extend this theory to the case of line con-

gruences {x, ξ}where xis a smooth map and ξis a proper frontal. For

these line congruences, we deﬁne the ﬁrst and second Kummer fundamental

forms associated to a moving basis Ω of the frontal ξ. Based on results of

[12], [10] and [11], we deﬁne the Ω-Kummer curvature function, denoted by

KΩ

q. Similarly to the case in which xis smooth, for each point p∈x(U),

the directions where this curvature assumes extreme values are the Kummer

principal directions. We determine the equation of the developable surfaces

and the equation of the principal surfaces of the congruence. Our main re-

sult is Theorem 4.1, in which we prove that for normal line congruences,

the equation of the principal surfaces is a multiple of the equation of the

developable surfaces, furthermore, the multiplicative factor is associated to

the singular set of ξ. As a corollary, we obtain the following extension of

Proposition 2.2: If xand ξare proper frontals with the same singular sets

and such that ξis the unit normal vector of x, then a curve on xis a direc-

trix curve of a principal surface of the congruence if and only if it is a line

of curvature of x.

The paper is organized as follows. In section 2 some basic concepts about

line congruences and Kummer’s theory are introduced. In section 3 notation

and some useful results about frontals are presented, taking into account the

approach used in [12]. Section 4 is addressed to the study of line congruences

{x, ξ}, where x:U→R3is a smooth map and ξ:U→R3is a proper frontal.

2. Background about line congruences

We now present basic concepts and properties of line congruences in R3.

The classical theory in R3has been given in [1], [4] and [17].

2.1. Kummer fundamental forms. In what follows, the space R3is ori-

ented by an once for all ﬁxed orientation and it is endowed with the Eu-

clidean inner product h,i.

Let C={x, ξ}be a line congruence in R3, where ξ, x :U⊂R2→R3are

smooth functions, kξk= 1 and ξis an immersion. The image S=x(U) is

called a reference set of the congruence. If xparametrizes a regular surface in

R3then Sis called reference surface. Let α:I⊂R→Ube a regular curve,

given by α(t) = (u1(t), u2(t)). Denote by x(t) = x(α(t)), ξ(t) = ξ(α(t)), q=

(u1(0), u2(0)), v=u0

1(0)xu1(q) + u0

2(0)xu2(q)=(u0

1(0), u0

2(0)) ∈TpS, where

p=x(q). Since ξis an immersion we could also consider w=u0

1(0)ξu1(q) +

LINE CONGRUENCES ON SINGULAR SURFACES 3

2(0)ξu2(q) in what follows, that is, we could take the tangent space of ξ

using the same coordinates (u0

1(0), u0

2(0)).

The following quadratic forms are associated to the congruence C.

(I) Kummer ﬁrst fundamental form :

Ip:TpS→R(1)

v7→ Ip(v) = Eu02

1+ 2Fu0

1u0

2+Gu02

where E=hξu1, ξu1i,F=hξu1, ξu2iand G=hξu2, ξu2i. We denote

by Ithe associated matrix.

(II) Kummer second fundamental form:

IIp:TpS→R(2)

v7→ IIp(v) = Lu02

1+ (M1+M2)u0

1u0

2+Nu02

where L=−hxu1, ξu1i,M2=−hxu1, ξu2i,M1=−hxu2, ξu1iand

N=−hxu2, ξu2i. We denote by II =−DξTDx the matrix of

these last coeﬃcients, where Dξ, Dx denote the Jacobian matrices

of ξ, x respectively.

In order to have Ipositive deﬁnite we suppose that ξis an immersion.

Usually, the Kummer second fundamental form is deﬁned with the opposite

sign (see [4] or [17]), but in this paper we work with the above deﬁnition.

Despite this, there are no changes on the geometry of the congruences. The

quadratic forms deﬁned above are called Kummer’s quadratic forms of the

congruence, because it was Kummer in Allgemeine Theorie der Gradlinigen

Strahen system who gave the ﬁrst purely mathematical treatment of line

congruences, see [4].

Deﬁnition 2.1. The lines of the congruence which pass through a curve

Con the reference surface Sform a ruled surface SCcalled surface of the

congruence.

If Cis given by x(α(t)) where α(t) = (u1(t), u2(t)) and ξ(t) = ξ(α(t)),

the surface of the congruence SCcan be written as

(3) Y(t, w) = x(t) + wξ(t), t ∈I, w ∈R,

where the curve α(t) is called a directrix of SCand for each ﬁxed tthe line

Lt, which pass through α(t) and is parallel to ξ(t), is called a generator

of the ruled surface SC. If ||ξ(t)|| = 1, we say that ξ(t) is the spherical

representation of SC. Suppose ||ξ(t)|| = 1 and ||ξ0(t)|| 6= 0. It is known (see

section 3.5 in [3]) that there exists a curve β:I→R3, contained in the

ruled surface SC, parametrized by

β(t) = x(t) + k(t)ξ(t),(4)

where k(t) = −hx0(t), ξ0(t)i

hξ0(t), ξ0(t)i, whose tangent vector satisﬁes

(5) hβ0(t), ξ0(t)i= 0.

LINE CONGRUENCES ON SINGULAR SURFACES 4

This special curve is called striction line. The intersection point of a genera-

tor with the striction line is called the central point of the generator. Given

a generator Ltthe coordinate of its central point is k(t), given in (4) . (See

ﬁgure 1)

Figure 1. Surface of the congruence SC, striction line βand

central point.

Let q= (u1(0), u2(0)) and note that

k(0) = −hx0(0), ξ0(0)i

hξ0(0), ξ0(0)i

=−hu0

1(0)xu1(q) + u0

2(0)xu2(q), u0

1(0)ξu1(q) + u0

2(0)ξu2(q)i

hu0

1(0)ξu1(q) + u0

2(0)ξu2(q), u0

1(0)ξu1(q) + u0

2(0)ξu2(q)i

=Lu02

1+ (M1+M2)u0

1u0

2+Nu02

Eu02

1+ 2Fu0

1u0

2+Gu02

=IIp

,where p=x(q).

If we associate to v=α0(0) = u0

1(0)xu1(q) + u0

2(0)xu2(q) its coordinates

(u0

1(0), u0

2(0)), then it is possible to look at kas a function deﬁned in TpS,

i.e

(6) Kp:TpS→R,Kp(v) = IIp(v)

Ip(v),

which gives the coordinate of the central point of the generator L0associated

to the surface of the congruence SC, determined by α, see ﬁgure 1. A point

pwhere the function Kpis constant is called a Kummer umbilic point.

The directions (du1, du2) where Kpassumes extreme values are called

Kummer principal directions . These directions are given by

(7) Lξdu2

1+Mξdu1du2+Nξdu2

2= 0,

where Lξ= 2F L −(M1+M2)E,Mξ= 2(G L −E N ) and Nξ=G(M1+

M2)−2F N (see section 95 in [17] for details).

If pis not a ξ-umbilic point then there exist two Kummer principal di-

rections l1and l2. The correspondent values of Kpin these directions,

K1=Kp(l1) and K2=Kp(l2), are called Kummer principal curvatures.

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LINECONGRUENCESONSINGULARSURFACESD.LOPESy;T.A.MEDINA-TEJEDA?;M.A.S.RUASzANDI.C.SANTOSAbstract.ThispaperisarststepinordertoextendKummer'sthe-oryforlinecongruencestothecasefx;g,wherex:U!R3isasmoothmapand:U!R3isaproperfrontal.Weshowthatiffx;gisanormalcongruence,theequationoftheprincipalsurfacesisa...

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LINE CONGRUENCES ON SINGULAR SURFACES D. LOPESyT.A. MEDINA-TEJEDAM.A.S. RUASzAND I.C. SANTOS Abstract. This paper is a rst step in order to extend Kummers the-

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