Surfaces with prescribed Weingarten­Operator 
Udo Simon \Lambda Konrad Voss Luc Vrancken y Martin Wiehe \Lambda 
Abstract 
We investigate pairs of surfaces in Euclidean 3­space with the same Weingarten 
operator in case that one surface is given as surface of revolution. Our local and 
global results complement global results on ovaloids of revolution from [S­V­W­W]. 
Keywords: Weingarten operator, surfaces of revolution 
2000 MSC: 53 A 05, 53 C 24, 53 C 40, 53 C 42 
Introduction 
In [S­V­W­W] we studied global uniqueness results of the following type: 
Theorem A. Let x; x # : M ! E 3 be ovaloids in Euclidean 3­space with nowhere dense 
umbilics and with the property that, at any p 2 M , the Weingarten operators S; S # and 
the spherical volume forms !(III), !(III # ) coincide: 
S = S # ; !(III) = !(III # ): 
Then x; x # are congruent. 
Corollary B. Let x; x # : M ! E 3 be ovaloids such that S = S # and !(III) = !(III # ). 
If x is analytic then x; x # are congruent up to a reparametrization. 
The basic idea for the proof of Theorem A is to consider the unique, I­selfadjoint, positive 
definite operator L defined by 
I # (v; w) =: I(Lv; Lw) 
and to study its algebraic and analytic properties. A second tool is to use the Codazzi 
equations for S = S # in terms of the two Levi­Civita connections r = r(I) and r # = 
r(I # ) to get relations for the symmetric (1,2) difference tensor (r 
\Lambda 
Partially supported by DFG. Both authors thank the deutsch­polnische Stiftung for partial support 
for the conference at the Banach Center in September 2000. 
y Supported by a research fellowship of the Alexander von Humbold Stiftung (Germany). 
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If one follows the proof it seems that one might drop the assumption on the volume forms. 
We would like to state the following 
Conjecture. Let x; x # : M ! E 3 be ovaloids with nowhere dense umbilics and with 
S = S # at corresponding points. Then x; x # are congruent. 
An affirmative answer to this conjecture would be an extrinsic counterpart to the intrinsic 
rigidity result of Cohn­Vossen [COHN­V] which states that two isometric ovaloids in E 3 
are congruent. In Section 4 of [S­V­W­W] we gave the following partial answer to our 
conjecture. 
Theorem C. Let x : M ! E 3 be an ovaloid of revolution with nowhere dense umbilics 
and let x # : M ! E 3 be another ovaloid with S = S # at corresponding points. Then 
x; x # are congruent. 
At the end of [S­V­W­W] we showed by an explicit example that there exists a non­ 
trivial 1­parameter family of complete surfaces of revolution x c : M ! E 3 having the 
same Weingarten operator. It is the aim of this paper (Sections 2--4) to give a complete 
discussion of the local situation of a pair of surfaces x; x # : M ! E 3 , where x is a surface 
of revolution and where the Weingarten operators coincide for any p 2 M . In Section 5 
we state the local results, in Section 6 we extend Theorem C. 
2 Surfaces of revolution 
We summarize well known properties of surfaces of revolution which we will need for our 
discussion. 
Consider a surface of revolution given in terms of parameters (u 1 ; u 2 ) by 
x(u 1 ; u 2 ) = (r(u 1 ) cos u 2 ; r(u 1 ) sin u 2 ; s(u 1 )); r – 0; (2.0.1) 
where u 1 parametrizes the meridians as arc length parameter and u 2 parametrizes the 
parallels of latitude with radius r(u 1 ) and r and s are differentiable functions. For a 
function f = f(u 1 ) we write f 0 := df=du 1 ; thus we have 
r 0 (u 1 ) 2 + s 0 (u 1 ) 2 = 1: (2.0.2) 
u 1 ; u 2 are curvature line parameters for all points with r(u 1 ) ? 0. In case that the 
functions r(u 1 ) ? 0, s(u 1 ) are defined for 0 ! u 1 ! \Lambda and also for u 1 = 0 or u 1 = \Lambda such 
that 
r(0) = 0; r 0 (0) = 1 or r(\Lambda) = 0; r 0 (\Lambda) = 
) sin u 2 are parameters for the 
surface. In general, the parameter u 2 will be taken modulo 2ß, i.e. u 2 2 S 1 ; but there 
will be cases where one has to pass to a covering surface by taking u 2 2 R. In every case, 
according to the domain of the parameters, there is a manifold M of dimension two such 
that (2.0.1) defines an immersion x : M ! E 3 . 
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It follows by a straightforward computation that the first fundamental form I =: g has 
the representation on MnfPN ; P S g: 
g 11 = 1; g 12 = 0; g 22 = r 2 ; 
and r and s satisfy 
S@ 1 = (r 0 s 00 
2 g denotes the Gauß basis associated to the local parameters. 
The equation (r 0 ) 2 + (s 0 ) 2 = 1 suggests to introduce the function oe = oe(u 1 ) by cos oe := r 0 
and sin oe := s 0 . The Weingarten operator is represented by the matrix 
S : 
0 
@ k 1 0 
0 k 2 
1 
A = 
0 
@ oe 0 0 
0 1 
r sin oe 
1 
A 
with k 1 ; k 2 as principal curvatures. The Codazzi equations reduce to the equation 
k 0 
2 = r 0 
r 
(k 1 
3 Pairs of surfaces with the same Weingarten opera­ 
tor 
In the beginning of this section we recall some local results from [S­V­W­W]. For a 
moment, let M be a connected, oriented C 1 ­manifold of dimension two; later again, 
we will restrict to M as given in Section 2. Let x; x # : M ! E 3 be a pair of surfaces 
with first fundamental forms g; g # and associated Levi­Civita connections r := r(g), 
r # := r(g # ) and Weingarten operators S = S # . We have the following facts from 
section 2 in [S­V­W­W]. 
3.1 Facts. 
(i) There exists a unique g­self­adjoint, positive definite operator L such that 
g # (u; v) = g(Lu; Lv) (3.1.1) 
for tangent vectors u; v. Denote the positive eigenvalue functions of L by – 1 ; – 2 ; 
they are continuous on M and, if – 1 6= – 2 , differentiable. 
(ii) The operator L and the Weingarten operator S = S # commute. 
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(iii) If x; x # admit a common curvature line parametrization on an open set U ae M , 
then the eigendirections of S are eigendirections of L, and we have the following 
local matrix representations: 
g : 
0 
@ g 11 0 
0 g 22 
1 
A ; g # : 
0 
@ – 2 
1 g 11 0 
0 – 2 
2 g 22 
1 
A ; 
L : 
0 
@ – 1 0 
0 – 2 
1 
A ; S = S # : 
0 
@ k 1 0 
0 k 2 
1 
A : 
(iv) If – 1 ; – 2 are differentiable, their partial derivatives in terms of curvature line pa­ 
rameters satisfy 
@ 2 – 1 = 0 = @ 1 – 2 : 
These equations are consequences of the Codazzi equations for S = S # in terms of 
r and r # , resp.; they are an essential tool for our discussion below. 
3.2 Codazzi equations. In curvature line parameters and with associated Gauß basis 
f@ 1 ; @ 2 g, the Codazzi equations for x read: 
2 @ 2 k 1 = 
4 The local discussion 
Let x : M ! E 3 be a surface of revolution as given in Section 2 and x # : M ! E 3 a 
second surface with the same parameter domain M . Let U ae M be open, connected and 
assume that x is without umbilics on U . Then we can apply the parametrization and 
matrix representations from 3.1(iii); moreover, – 1 and – 2 are differentiable on U . 
We discuss the consequences of the integrability conditions in terms of the representations 
in 3.1(iii). One easily verifies that there is no further information from the Codazzi 
equations. The Gauß equation and K = det S = det S # = K # lead to an ODE which is 
crucial for our discussion; see Section 4 in [S­V­W­W]. 
4.1 Proposition. The functions r = r(u 1 ) and ø := (– 1 ) 
R such that 
(r 0 ) 2 ø = c: (4.1.1) 
4.2 Consequences. 
(i) If c = 0, then r = const or – 1 = 1. 
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(ii) If c 6= 0, then r 0 6= 0 and 
– 1 = jr 0 j 
p 
(r 0 ) 2 + c 
6= 1: 
4.3 Discussion of the case c = 0. 
(a) If r =: r 0 = const then (r 0 ) 2 + (s 0 ) 2 = 1 implies s = u 1 and x lies on a circular 
cylinder 
x = (r 0 cos u 2 ; r 0 sin u 2 ; u 1 ): 
Choosing arbitrary positive functions – 1 (u 1 ); – 2 (u 2 ), the invariants g # and S # are 
given by 3.1 (iii); thus the surface x # is uniquely determined. 
Reparametrize 
¯ 
u 1 := 
Z u 1 
– 1 (u)du; ¯ 
u 2 := 
Z u 2 
– 2 (v)dv: (4.3.1) 
One verifies that the solution x # is given by 
x # = (r 0 cos ¯ 
u 2 ; r 0 sin ¯ 
u 2 ; ¯ 
u 1 ); 
thus x # is another parametrization of the same circular cylinder and the diffeomor­ 
phism (u 1 ; u 2 ) 7! (¯u 1 ; ¯ u 2 ) preserves the Weingarten operator: S = S # . 
(b) If – 1 = 1 we again reparametrize, using (4.3.1). Analogously to the foregoing we 
arrive at 
x # = (r(u 1 ) cos ¯ 
u 2 ; r(u 1 ) sin(¯u 2 ); s(u 1 )): (4.3.2) 
x # is a modified parametrization of the same surface of revolution. Again the 
diffeomorphism (u 1 ; u 2 ) 7! (u 1 ; ¯ u 2 ) preserves the Weingarten operator. 
4.4 Discussion of the case c 6= 0. For fixed c, the function – 1 is given by 4.2(ii); we 
reparametrize x # 
c by 
¯ 
u 1 
c := 
R u 1 
– 1 (u)du; ¯ u 2 
c := a 
R u 2 
– 2 (v)dv; 0 ! 
p 
1 + c =: a 2 R: (4.4.1) 
Introduce the functions 
¯ 
r(¯u 1 
c ) := 1 
a r(u 1 ) and ¯ 
s(¯u 1 ) := 1 
a 
Z u 1 p 
(a– 1 ) 2 
c, the surface x # 
c is given by 
x # 
c (¯u 1 ; ¯ u 2 ) = (¯r(¯u 1 ) cos ¯ 
u 2 ; ¯ r(¯u 1 ) sin ¯ u 2 ; ¯ s(¯u 1 )); 
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in particular, x # 
c is a surface of revolution; moreover S = S # 
c in corresponding points 
(u 1 ; u 2 ) 7! (¯u 1 
c ; ¯ 
u 2 
c ) for any c 6= 0. x # 
c is also defined for c = 0; the choice – 2 = a 
4.6 Remark. (a) Recall (r 0 ) 2 Ÿ 1 from (2.0.2). According to 4.2(ii) the inequality c 6= 0 
implies – 1 6= 1, in particular: 
(i) c ? 0 , – 1 ! 1; in this case we have – 1 Ÿ 1 
p 
1+c 
; 
(ii) c ! 0 , – 1 ? 1; in this case we have 
for (4.1.1). Integrating 
(4.1.1), we get r = – 1 
p 
c(1 
1 = u 1 
r 
c 
1 
1 
= – 1 u 1 
r c 
(1 + c)(1 
non­congruent). Especially s 
r 6= s # 
r # and ¯ u 2 = 
p 
1+c 
–1 u 1 6= 
u 1 since – 1 6= 
p 
1 + c. 
(c) In [S­V­W­W] we illustrated Proposition 4.5 by the example of the elliptic paraboloid 
x(u; v) = (u cos v; u sin v; 1 
2 u 2 ) 
having the same Weingarten operator as any surface of the one­parameter family of 
strongly convex surfaces of revolution 
x c (u; v) = ( u 
p 
1 + c 
cos v; 
u 
p 
1 + c 
sin v; 
1 
c 
[ 
r 
1 + c 
u 2 
1 + c 
? 0 and the surfaces x c are complete. For 
Section we state a series of consequences from the local discussion in Section 4. 
As before, x : M ! E 3 is a surface of revolution with the representation 
x(u 1 ; u 2 ) = (r(u 1 ) cos u 2 ; r(u 1 ) sin u 2 ; s(u 1 )) 
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on M ; we assume that the umbilics are nowhere dense. Moreover, let x # : M ! E 3 be a 
surface with S # = S. 
The detailed discussion in Section 4 admits the following implications. 
5.1 Let x; x # : M ! E 3 be given as before; then there exist coordinates ¯ 
u 1 ; ¯ 
u 2 in M , 
and differentiable functions ¯ r = ¯ 
r(¯u 1 ); ¯ s = ¯ s(¯u 1 ), such that x # has the representation 
(modulo congruences in E 3 ): 
x # (¯u 1 ; ¯ u 2 ) = (¯r(¯u 1 ) cos ¯ 
u 2 ; ¯ r(¯u 1 ) sin ¯ u 2 ; ¯ s(¯u 1 )); 
in particular, x # again is a surface of revolution. 
Proof. Sections 4.2 ­ 4.6 give a discussion of all possible cases. 
5.2 There exists a non­trivial one­parameter family of strongly convex surfaces of revo­ 
lution x c : M ! E 3 having the same Weingarten operator at corresponding points. For 
c ? 0, the examples in 4.6(c) are complete and admit a bijective orthogonal mapping onto 
the plane (x 1 ; x 2 ; 0) ae R 3 . 
5.3 There exist non­isometric two­parameter families x = x c;–1 , x # = x # 
c;–1 of surfaces 
of revolution (see 4.6(b)) with the properties 
(i) the surfaces x; x # , have the same Weingarten operator and nowhere dense umbilics; 
(ii) the surfaces x; x # have the same Riemannian volume. 
Such surfaces necessarily are circular cones. 
Proof. Almost everywhere we can introduce the local parameters from 3.1. (ii) implies 
1 = det L = – 1 \Delta – 2 which together with @ 2 – 1 = 0 = @ 1 – 2 gives – 1 = const 6= 1; – 2 = 
const 6= 1 (as x; x # are assumed to be non­isometric). Then c 6= 0 in (4.1.1), and 4.6(b) 
describes the solution. 
Remark. The foregoing result in particular implies that both metrics g; g # must be 
flat. Thus any pair x; x # : M ! E 3 , where x is a non­flat surface of revolution with 
nowhere dense umbilics which satisfies (i) and (ii) in 5.3, must be congruent. 
5.4 Let x; x # be given as in the beginning of this section. Let p 2 M be a pole for x. 
Then p is also a pole for the surface of revolution x # . Using 3.1(i), we have – 1 = – 2 at 
p. 
Proof. 4.3(b) or 4.4; other cases are excluded. 
5.5 Corollary. Let D ae M be a geodesic disc for x (open or closed) with the pole p as 
center; then – 2 = const on D. 
Proof. See the proof of Theorem 4.1 in [S­V­W­W]. 
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5.6 Corollary. Assume r 6= const and let the disc D from Corollary 5.5 contain a point 
(and then a parallel of latitude) with the property r 0 (p) = 0 (or lim 
q!p 
r 0 (q) = 0). Then x 
and x # are congruent on D. 
Proof. Proposition 4.1 and Corollary 5.5 imply 1 = – 1 = – 2 and thus g = g # . 
6 Global results 
In Section 4 of [S­V­W­W] we considered ovaloids of revolution x; x # : M ! E 3 with 
nowhere dense umbilics and S = S # . We proved that x; x # must be congruent (see 
Theorem C). The comments in 5.2 show that, for a rigidity result, one cannot weaken 
the assumptions and consider x to be a complete convex surface of revolution instead. 
Nevertheless, we can generalize Theorem C as follows. 
6.1 Theorem. Let M be a surface of genus zero and x; x # : M ! E 3 be immersions. 
As before assume that x is a surface of revolution as given in (2.0.1) having nowhere 
dense umbilics and r(u 1 ) 6= const on open nonempty sets. Then S = S # implies the 
congruence of x and x # . 
Proof. Genus(M ) = 0 implies that the curve u 1 7! (r(u 1 ); s(u 1 )) is an arc with r(u 1 ) ? 0 
for 0 ! u 1 ! \Lambda and r(0) = r(\Lambda) = 0; thus r 0 has a zero between 0 and \Lambda, and Corollaries 
5.5 and 5.6 imply – 1 = – 2 = 1 on M . 
References 
[COHN­V] Cohn­Vossen, S.: Zwei S¨atze ¨uber die Starrheit der Eifl¨achen. Nachr. der Ges. der 
Wissensch. zu G¨ottingen, Math.­Phys. Kl., Jahrg. 1927, 125--134. 
[S­V­W­W] Simon, U.; Vrancken, L.; Wang, C.P.; Wiehe, M.: Intrinsic and extrinsic geometry of 
ovaloids and rigidity, Geometry and Topology of Submanifolds X, World Scientific, 
Singapore, 284­293, 2000. 
U. Simon and M. Wiehe, Fak.II, Institut Mathematik, MA 8­3, TU­Berlin, Germany. 
K. Voss, Department Mathematik, ETH­Zentrum, Z¨urich, Switzerland. 
L. Vrancken, Department Mathematics, University Utrecht, The Netherlands. 
e­mail: simon@math.tu­berlin.de 
voss@math.ethz.ch 
vrancken@math.uu.nl 
wiehe@math.tu­berlin.de 
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