Properties and Transformations of Weingarten Surfaces Brendan Guilfoyle1and Morgan Robson12

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Properties and Transformations

of Weingarten Surfaces

Brendan Guilfoyle1,* and Morgan Robson1,2,†

1School of STEM, Munster Technological University, Tralee, Co. Kerry, Ireland.

2Department of Computing and Mathematics, South East Technological University,

Waterford, Ireland.

*brendan.guilfoyle@mtu.ie

†Corresponding Author: morgan.robson@setu.ie

December 5, 2024

Abstract

This paper explores Weingarten relations satisﬁed by surfaces of revolution in Euclidean

3-space E3. Firstly, we establish that the local geometry of a surface around umbilic points

restricts its possible Weingarten relations. We demonstrate that the rate at which the surface

becomes spherical at umbilic points imposes bounds on the slope of any satisﬁed Weingarten

relation, extending previous research by a number of authors.

Secondly, we investigate transformations between Weingarten relations through the action

of SL2(R), acting as fractional linear transformations on the surface’s curvatures. We integrate

this action, which splits into three natural geometric actions on surfaces in E3, providing a

method of generating rotationally symmetric solutions to a transformed Weingarten relation.

This technique is applied to a class of Weingarten relations known as semi-quadratic. We prove

the action is transitive on such relations and give a classiﬁcation result on their solutions.

1 Introduction

Introduced by J. Weingarten in 1861 [30], Weingarten surfaces are a topic of classical diﬀerential

geometry and have found applications in architectural design [26,27,29]. An oriented surface in

Euclidean 3-space E3is Weingarten when its principal curvatures k1and k2satisfy a diﬀerentiable

functional relationship expressed as

W(k1, k2)=0.(1.1)

Wis called the Weingarten relationship and is a non-linear second-order PDE satisﬁed by the

surface. Recent work has focused on understanding how Weingarten relations determine geometric

2010 Mathematics Subject Classiﬁcation: 53A05, 53C42

Keywords: Weingarten surface, rotational symmetry, curvature, group action

arXiv:2210.10035v4 [math.DG] 4 Dec 2024

properties of their rotationally symmetric solutions [3,5]. In conjunction with this theme, this paper

investigates the possible Weingarten relations for surfaces of revolution. This is done through two

approaches. Firstly, obstruction criteria for Weingarten relations are given in terms of local surface

geometry around umbilic points. Secondly, given an initial surface of revolution and its Weingarten

relation, a family surface transformations are applied, generating solutions to transformed Wein-

garten relationships. A classiﬁcation result of certain Weingarten surfaces is then given. Further

details are now given.

Obstructions to Weingarten Relations for Surfaces of Revolution

Our ﬁrst topic explores how the geometric behaviour of a surface near its umbilic points aﬀects the

supported Weingarten relations. This is done in terms of a surface’s curvature diagram, denoted

as F(S), which represents the set of curvatures (k1, k2) attained by points on the surface Sas a

subset of the k1k2-plane. In the literature, k1and k2are typically labelled by the condition k2≤k1.

A surface Sis Weingarten with relation (1.1) if and only if F(S)⊆W−1{0}, hence F(S) strongly

determines the supported Weingarten relationships. Examples are depicted in Figure 1. Points of S

with equal principal curvatures are called umbilic points and the diagonal k1=k2in the k1k2-plane

is called the umbilic axis. The study of umbilic points is a classical yet still active area of research

[8,13,14]. They are guaranteed to exist on closed C2surfaces of zero genus, and thus the curvature

diagram of such surfaces must intersect the umbilic axis.

F(S)

Umbilic Axis

F(S)

Umbilic Axis

Figure 1: Left: Pairs (k1, k2) satisfying a CMC relationship k1+k2=c(orange) and the curvature

diagrams of a generic surface (blue) and a generic Weingarten surface (red). Right: The directions

of negative slope from the umbilic axis.

Various authors have described the possible shapes of curvature diagrams, near the umbilic axis

[4,6,16,19]. Diﬀerent assumptions on the surface Sare made, see [4], Lemma 2 or [6], Theorem

1.1 for examples, however the general conclusion is that if F(S) intersects the umbilic axis, it does

so with a non-negative slope in the k1k2-plane, or, is a point on the umbilic axis (and thus Sis con-

gruent to a subset of the round sphere or a plane). Directions of negative slope are shaded in Figure

1. Due to the above, the orange curve in Figure 1cannot be the curvature diagram of a surface,

as it meets the umbilic axis from a direction of negative slope. Thus surfaces congruent to round

spheres are the only surface homeomorphic to S2which can satisfy the relationship k1+k2=c,

c > 0, as is well known [1].

In this paper we give strictly positive lower bounds on the slope at which F(S) intersects the

umbilic axis for SaC2-smooth surface of revolution. Throughout the rest of the paper, Swill

denote such a surface, in which case F(S) is generically a curve (being the continuous image of the

proﬁle curve of S).

•If p∈ S is an umbilic point, the umbilic slope at p, denoted µp, is the slope at which F(S)

meets the umbilic axis. We remark that µpmay not be well deﬁned for every S.

•A surface is said to be totally umbilic around pif there exists a neighbourhood of pin Swhich

contains only umbilic points.

•Swill be called non-ﬂat at the point q∈ S if K(q)̸= 0, and non-ﬂat if it is non-ﬂat at every

point.

•Swill be called convex at the point q∈ S if K(q)≥0, and convex if it is convex at every

point.

•Swill be called strictly convex at the point q∈ S if K(q)>0, and strictly convex if it is

strictly convex at every point.

We remark that our usage of the word ‘non-ﬂat’ here does not coincide with the idea of a surface

being distinct from a plane, locally. A point is non-ﬂat if and only if it is not a parabolic and planar

point.

We ﬁrst consider an umbilics points which lie oﬀ the axis of rotational symmetry. With a mi-

nor technical assumption on the umbilic p, we show that if Sis non-ﬂat at p, then if F(S) has a

tangent line at p, it must be vertical. (Theorem 3.2). The behaviour of µpfor an umbilic point p

lying on the axis of rotational symmetry is then considered. Let r1and r2be the radii of curvature

of Sand θthe angle formed between S’s (oriented) axis of rotational symmetry and its oriented

normal vector. We may assume without loss of generality that θ= 0 at p.

Theorem 3.8. Let Sbe strictly convex at an isolated umbilic point pon the axis of rotational

symmetry. Suppose at p,Shas an umbilic slope of µp∈R.

(A) If the radii of curvature satisfy lim

θ→0r2−r1

sinαθ=γfor some α≥0,γ∈R, then µp≥α+ 1, with

equality if γ̸= 0.

(B) Conversely if µp> α + 1 then lim

θ→0r2−r1

sinαθ= 0.

Theorem 3.8 is a corollary of Theorem 3.7 which bounds the limit superior and limit inferior of the

average slope of R(S) near an umbilic point, covering cases where µpmay not be well deﬁned. As

a corollary of Theorem 3.8 we show the following

Corollary 3.10. Let pbe an isolated umbilic point on the axis of rotational symmetry of S, a

strictly convex and C3-smooth surface. Then if µpexists, µp≥2. If in addition Sis C4-smooth

then µp≥3.

The above theorems hence characterise µpas a measure of the rate at which Sbecomes umbilic.

SL2(R)Transformations

Our second topic concerns a family of transformations which, for each Weingarten relation (1.1)

having a rotationally symmetric solution, produces a new relation which also admits a rotationally

symmetric solution. SL2(R) acts on the k1k2-plane by real fractional linear transformations, which

coincide with the isometries of the geometrised k1k2-plane considered in [10]. Our main theorem is

Theorem 4.8. If Tis a real fractional linear transformation of the k1k2-plane with Snon-ﬂat, then

there exists a rotationally symmetric and possibly non-regular surface e

Ssuch that F(e

S) = T(F(S)).

The action of SL2(R) is then described geometrically in terms of transformations of Sin E3(The-

orem 4.14). A class of surfaces called semi-quadratic Weingarten surfaces satisfying a Weingarten

relation of the form

αk1k2+βk1+γk2+δ= 0 α, β, γ, δ ∈R,(1.2)

are investigated. This class contains well-known subclasses of surfaces, such as ones linear in k1and

k2, ones linear in the mean and Gauss curvature, Hand K, and ones linear in the radii of curvature,

r1=1

k1and r2=1

k2, investigated in [22,24], [7,23] and [12], respectively. The quantities

Λ1=β−γ, Λ2= (β+γ)2−4αδ,

are introduced which characterise when the PDE (1.2) is elliptic, namely Λ2>Λ2

1(Proposition 4.19).

When Λ1= 0 semi-quadratic surfaces become LW-surfaces. LW-surfaces are classiﬁed into three

types, elliptic when Λ2>0 [7], hyperbolic when Λ2<0 [23] and a border case Λ2= 0 which describe

subsets of spheres, tubular surfaces or planes. This motivates a generalisation of the nomenclature:

Deﬁnition 1.1. A semi-quadratic Weingarten surface satisfying Λ2>Λ2

1is said to be elliptic. If

Λ2<Λ2

1it is said to be hyperbolic.

Semi-quadratic relations form an invariant set under the action of SL2(R) and the ratio Λ2

1/Λ2

is shown to be an invariant (Proposition 4.21). The SL2(R) transformations are then shown to

be transitive on all semi-quadratic relations satisfying Λ2>0 and sharing the same invariant

(Proposition 4.23), therefore such semi-quadratic surfaces can be transitively related by induced

transformations in E3. This is used to show the following.

Theorem 4.24. Let Sbe a connected rotationally symmetric semi-quadratic Weingarten surface

for which Λ2

1= Λ2. Then Sis a subset of a round sphere, tubular surface or plane.

Theorem 4.28. Any non-ﬂat rotationally symmetric, connected semi-quadratic Weingarten surface

with Λ2>0is the image under a composition of homotheties, parallel translations and reciprocal

transformations of a Weingarten surface satisfying the relation

k2=λk1,(1.3)

for λ > 0when the surface is elliptic, or for λ < 0when the surface is hyperbolic.

Surfaces of revolution satisfying relation (1.3) were classiﬁed in [24]. Theorem 4.28 therefore ex-

tends this classiﬁcation to rotationally symmetric semi-quadratic surfaces with Λ2>0.

The paper is organised as follows. Section 2details our method of describing surfaces of revolu-

tion and introduces the radius of curvature equivalent of the curvature diagram, termed the RoC

diagram. Section 3explores the possible RoC diagrams for surfaces of revolution while Section 4

investigates the eﬀect of the SL2(R) mappings on Weingarten relations and the transformations

they induce on surfaces.

2 Background

2.1 The Curvature of Surfaces of Revolution.

In this short subsection we deﬁne a coordinate system on Sin the special case Sis non-ﬂat. By

continuity of the Gauss curvature, if a C2-smooth surface is non-ﬂat at a point it is also non-ﬂat in

a neighbourhood of that point - hence the constructed coordinates will be used to describe surfaces

of revolution locally around non-ﬂat points. Position Sin R3with the axis of rotational symmetry

aligned with the zaxis. The principal foliations of Sare given by the parallels and proﬁle curves

of Swhose respective principal curvatures we denote by k1and k2. Note because Sis assumed

non-ﬂat, both k1and k2are non-zero. Let αbe the proﬁle curve of Swhich lies in the yz-plane

(i.e. the generating curve of S). Since k2̸= 0 the Gauss map N:α→S1is a local diﬀeomorphism

and we may thus use the Gauss angle θ∈(−π, π] on S1to locally parameterise α. Here θis the

angle made between the normal vector of α, denoted ˆn, and the positive zaxis. We orient αso that

ˆn(θ) = (sin(θ),cos(θ)). Points of αon the zaxis such that θ= 0 or θ=πwill be called north and

south poles respectively. If ρand hdenote the respective yand zcomponents of α, they satisfy the

relationship dh

dρ=−tan θ, (2.1)

and Smay be described via ⃗

X(θ, ϕ)=(ρ(θ) sin ϕ, ρ(θ) cos ϕ, h(θ)) for (θ, ϕ)∈(−π, π]×[0, π). This

is illustrated in Figure 2.

ˆn

Figure 2: The proﬁle curve α(blue) in the yz-plane is parameterised by the angle θand revolved

by an angle of ϕaround the axis of rotation (black) to generate S.

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PropertiesandTransformationsofWeingartenSurfacesBrendanGuilfoyle1,*andMorganRobson1,2,†1SchoolofSTEM,MunsterTechnologicalUniversity,Tralee,Co.Kerry,Ireland.2DepartmentofComputingandMathematics,SouthEastTechnologicalUniversity,Waterford,Ireland.*brendan.guilfoyle@mtu.ie†CorrespondingAuthor:morgan.rob...

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Properties and Transformations of Weingarten Surfaces Brendan Guilfoyle1and Morgan Robson12

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