# An Intrinsic Characterization of Bonnet Surfaces Based on a Closed Differential Ideal

Authors: Paul Bracken
Journal: Geometry
 (2014)

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## Abstract

The structure equations for a two-dimensional manifold are introduced and two results based on the Codazzi equations pertinent to the study of isometric surfaces are obtained from them. Important theorems pertaining to isometric surfaces are stated and a theorem due to Bonnet is obtained. A transformation for the connection forms is developed. It is proved that the angle of deformation must be harmonic, and that the differentials of many of the important variables generate a closed differential ideal. This implies that a coordinate system exists in which many of the variables satisfy particular ordinary differential equations, and these results can be used to characterize Bonnet surfaces.


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## Body

## 1. Introduction

Bonnet surfaces in three-dimensional Euclidean space have been of great interest for a number of reasons as a type of surface [1, 2] for a long time. Bonnet surfaces are of nonconstant mean curvature that admit infinitely many nontrivial and geometrically distinct isometries which preserve the mean curvature function. Nontrivial isometries are ones that do not extend to isometries of the whole space E3. Considerable interest has resulted from the fact that the differential equations that describe the Gauss equations are classified by the type of related Painlevé equations they correspond to and they are integrated in terms of certain hypergeometric transcendents [3–5]. Here the approach first given by Chern [6] to Bonnet surfaces is considered. The developement is accessible with many new proofs given. The main intention is to end by deriving an intrinsic characterization of these surfaces which indicates they are analytic. Moreover, it is shown that a type of Lax pair can be given for these surfaces and integrated. Several of the more important functions such as the mean curvature are seen to satisfy nontrivial ordinary differential equations.Quite a lot is known about these surfaces. With many results the analysis is local and takes place under the assumptions that the surfaces contain no umbilic points and no critical points of the mean curvature function. The approach here allows the elimination of many assumptions and it is found that the results are not too different from the known local ones. The statements and proofs have been given in great detail in order to help illustrate and display the interconnectedness of the ideas and results involved.To establish some information about what is known, consider an oriented, connected, smooth open surfaceM in E3 with nonconstant mean curvature function H. Moreover, M admits infinitely many nontrivial and geometrically distinct isometries preserving H. Suppose that U is the set of umbilic points of M and V is the set of critical points of H. Many global facts are known with regard to U, V, and H, and a few will now be mentioned. The set U consists of isolated points, even if there exists only one nontrivial isometry preserving the mean curvature; moreover, U⊂V [7, 8]. Interestingly, there is no point in V-U at which all order derivatives of H are zero, and V cannot contain any curve segment. If the function by which a nontrivial isometry preserving the mean curvature rotates the principal frame is considered, as when there are infinitely many isometries, this function is a global function on M continuously defined [9–11]. As first noted by Chern [6], this function is harmonic. The analysis will begin by formulating the structure equations for two-dimensional manifolds.


## 2. Structure Equations

OverM there exists a well defined field of orthonormal frames which is written as x, e1, e2, e3 such that x∈M, e3 is the unit normal at x and e1, e2 are along principal directions [12]. The fundamental equations for M have the form
(1)dx=ω1e1+ω2e2,de1=ω12e2+ω13e3,de2=-ω12e1+ω23e3,de3=-ω13e1-ω23e2.
Differentiating each of these equations in turn, results in a large system of equations for the exterior derivatives of the ωi and the ωij, as well as a final equation which relates some of the forms [13]. This choice of frame and Cartan’s lemma allows for the introduction of the two principal curvatures which are denoted by a and c at x by writing
(2)ω12=hω1+kω2,ω13=aω1,ω23=cω2.
Suppose that a>c in the following. The mean curvature of M is denoted by H and the Gaussian curvature by K. They are related to a and c as follows:
(3)H=12(a+c),K=a·c.
The forms which appear in (1) satisfy the fundamental structure equations which are summarized here [14],
(4)dω1=ω12∧ω2,dω2=ω1∧ω12,dω13=ω12∧ω23,dω23=ω13∧ω12,dω12=acω2∧ω1=-Kω1∧ω2.
The second pair of equations of (4) is referred to as the Codazzi equation and the last equation is the Gauss equation.Exterior differentiation of the two Codazzi equations yields(5)(da-(a-c)hω2)∧ω1=0,(dc-(a-c)kω1)∧ω2=0.
Cartan’s lemma can be applied to the equations in (5). Thus there exist two functions u and v such that
(6)1a-cda-hω2=(u-k)ω1,1a-cdc-kω1=(v-h)ω2.
Subtracting the pair of equations in (6) gives an expression for dlog⁡(a-c)(7)dlog⁡(a-c)=(u-2k)ω1-(v-2h)ω2.
Define the variable J to be
(8)J=12(a-c)>0.
It will appear frequently in what follows. Equation (7) then takes the form
(9)dlog⁡J=(u-2k)ω1-(v-2h)ω2.
The ωi constitute a linearly independent set. Two related coframes called ϑi and αi can be defined in terms of the ωi and the functions u and v as follows:
(10)ϑ1=uω1+vω2,ϑ2=-vω1+uω2,α1=uω1-vω2,α2=vω1+uω2.
These relations imply that ϑ1=0 is tangent to the level curves specified by H equals constant and α1=0 is its symmetry with respect to the principal directions.Squaring both sides of the relation2H=a+c and subtracting the relation 4K=4ac yields 4(H2-K)=(a-c)2. The Hodge operator, denoted by *, will play an important role throughout. It produces the following result on the basis forms ωi,
(11)*ω1=ω2,*ω2=-ω1,*2=-1.
Moreover, adding the expressions for da and dc given in (6), and the results are
(12)1a-c(da+dc)=(u-k)ω1+hω2+(v-k)ω2+kω1=uω1+vω2=ϑ1.
Finally, note that
(13)α1+2*ω12=uω1-vω2+2*(hω1+kω2)=(u-2k)ω1-(v-2h)ω2=dlog⁡J.
Therefore, the Codazzi equations (12) and (13) can be summarized using the definitions of H and J as
(14)dH=Jϑ1,dlog⁡J=α1+2*ω12.


## 3. A Theorem of Bonnet

Suppose thatM* is a surface which is isometric to M such that the principal curvatures are preserved [10–12]. Denote all quantities which pertain to M* with the same symbols but with asterisks, as for example,
(15)a*=a,c*=c.
The same notation will be applied to the variables and forms which pertain to M and M*. When M and M* are isometric, the forms ωi are related to the ωi* by the following transformation
(16)ω1*=cos⁡τω1-sinτω2,ω2*=sinτω1+cos⁡τω2.Theorem 1.
Under the transformation of coframe given by (16), the associated connection forms are related by
(17)ω12*=ω12-dτ.Proof.
Exterior differentiation ofω1* produces
(18)dω1*=-sinτdτ∧ω1+cos⁡τdω1-cos⁡τdτ∧ω2-sinτdω2=dτ∧(-sinτω1-cos⁡τω2)+cos⁡τω12∧ω2-sinτω1∧ω12=(-dτ+ω12)∧ω2*.
Similarly, differentiating ω2* gives
(19)dω2*=cos⁡τ∧ω1+sinτdω1-sinτdτ∧ω2+cos⁡τdω2=dτ∧(cos⁡τω1-sinτω2)+sinτω12∧ω2+cos⁡τω1∧ω12=ω1*∧(-dτ+ω12).There is a very important result which can be developed at this point. In the case thata=a* and c=c*, the Codazzi equations imply that
(20)α1+2*ω12=dlog⁡(a-c)=dlog⁡(a*-c*)=α1*+2*ω12*.
Apply the operator * to both sides of this equation, we obtain
(21)α2-2ω12=α2*-2ω12*.
Substituting for ω12* from Theorem 1, this is
(22)2dτ=α2-α2*.Lemma 2.
Consider(23)ϑ1=ϑ1*.Proof.
This can be shown in two ways. First from (16), express the ωi in terms of the ωi* to give
(24)ω1=cos⁡τω1*+sinτω2*,ω2=-sinτω1*+cos⁡τω2*.
Therefore,
(25)ϑ1=uω1+vω2=u(cos⁡τω1*+sinτω2*)+v(-sinτω1*+cos⁡τω2*)=u*ω1*+v*ω2*=ϑ1*,
where u*=ucos⁡τ-vsinτ and v*=usinτ+vcos⁡τ.Lemma2 also follows from the fact that dH=dH* and (8).Lemma 3.
Consider(26)α2*=sin(2τ)α1+cos⁡(2τ)α2.Proof.
(27)
α
2
*
=
(
u
sin
τ
+
v
cos
⁡
τ
)
(
cos
⁡
τ
ω
1
-
sin
τ
ω
2
)
+
(
u
cos
⁡
τ
-
v
sin
τ
)
(
sin
τ
ω
1
+
cos
⁡
τ
ω
2
)
=
(
u
sin
(
2
τ
)
+
v
cos
⁡
(
2
τ
)
)
ω
1
+
(
-
v
sin
(
2
τ
)
+
u
cos
⁡
(
2
τ
)
)
ω
2
=
sin
(
2
τ
)
α
1
+
cos
⁡
(
2
τ
)
α
2
.Substitutingα2* from Lemma 3 into (22), dτ can be written as
(28)dτ=12(α2-sin(2τ)α1-cos⁡(2τ)α2)=12((1-cos⁡(2τ))α2-sin(2τ)α1).
Introduce the new variable t=cot(τ) so dt=-csc2(τ)dτ and sinτ=1/1+t2, cos⁡τ=1/1+t2, hence the following lemma.Lemma 4.
Consider(29)dt=tα1-α2.This is the total differential equation which must be satisfied by the angleτ of rotation of the principal directions during the deformation. If the deformation is to be nontrivial, it must be that this equation is completely integrable.Theorem 5.
A surfaceM admits a nontrivial isometric deformation that keeps the principal curvatures fixed if and only if
(30)dα1=0,dα2=α1∧α2
or α12=α2.Proof.
Differentiating both sides of Lemma4 gives
(31)dt∧α1+tdα1-dα2=(tα1-α2)∧α1+tdα1-dα2=0.
Equating the coefficients of t to zero gives the result (30).This theorem seems to originate with Chern [6] and is very useful because it gives the exterior derivatives of the αi. When the mean curvature is constant, dH=0, hence it follows from (14) that ϑ1=0. This implies that u=v=0, and so α1 and α2 must vanish. Hence dt=0 which implies that, since the αi are linearly independent, t equals a constant. Thus, we arrive at a theorem originally due to Bonnet.Theorem 6.
A surface of constant mean curvature can be isometrically deformed preserving the principal curvatures. During the deformation, the principal directions rotate by a fixed angle.


## 4. Connection Form Associated to a Coframe and Transformation Properties

Given the linearly independent one-formsω1, ω2, the first two of the structure equations uniquely determine the form ω12. The ω1, ω2 are called the orthonormal coframe of the metric
(32)ds2=ω12+ω22,
and ω12 the connection form associated with it.Theorem 7.
Suppose thatA>0 is a function on M. Under the change of coframe,
(33)ω1*=Aω1,ω2*=Aω2,
the associated connection forms are related by
(34)ω12*=ω12+*dlog⁡A.Proof.
The structure equations for the transformed system are given as(35)dω1*=ω12*∧ω2*,dω2*=ω1*∧ω12*.
Using (33) to replace the ωi* in these, we obtain
(36)dlog⁡A∧ω1+dω1=ω12*∧ω2,dlog⁡A∧ω2+dω2=ω1∧ω12*.
The set of ωi satisfies a similar system of structure equations, so replacing dωi here yields
(37)(ω12*-ω12)∧ω2=dlog⁡A∧ω1,(ω12*-ω12)∧ω1=-dlog⁡A∧ω2.
Since the forms ωi satisfy the equations *ω1=ω2 and *ω2=-ω1, substituting these relations into the above equations and using Ωk∧(*Θk)=Θk∧(*Ωk), we obtain that in the form
(38)ω1∧*(ω12*-ω12)=-ω1∧dlog⁡A,ω2∧*(ω12*-ω12)=-ω2∧dlog⁡A.
Cartan’s lemma can be used to conclude from these that there exist functions f and g such that
(39)*(ω12*-ω12)=-dlog⁡A-fω1,*(ω12*-ω12)=-dlog⁡A+gω2.
Finally, apply * to both sides and use *2=-1 to obtain
(40)ω12*-ω12=*dlog⁡A+fω2,ω12*-ω12=*dlog⁡A+gω1.
The forms ωi are linearly independent, so for these two equations to be compatible, it suffices to put f=g=0, and the result follows.For the necessity in the Chern criterion, Theorem5, no mention of the set V of critical points of H is needed. In fact, when H is constant, this criterion is met and the sufficiency also holds with τ constant. However, when H is not identically constant, we need to take the set V of critical points into account for the sufficiency. In this case, M-V is also an open, dense, and connected subset of M. On this subset J>0 and the function A can be defined in terms of the functions u and v as
(41)A=+u2+v2>0.
To define more general transformations of the ωi, define the angle ψ as
(42)u=Acos⁡(ψ),v=Asin(ψ).
This angle which is defined modulo 2π, is continuous only locally and could be discontinuous in a nonsimply connected region of M-V. With A and ψ related to u and v by (42), the forms ϑi and αi can be written in terms of A and ψ as
(43)ϑ1=A(cos⁡(ψ)ω1+sin(ψ)ω2),ϑ2=A(-sin(ψ)ω1+cos⁡(ψ)ω2),α1=A(cos⁡(ψ)ω1-sin(ψ)ω2),α2=A(sin(ψ)ω1+cos⁡(ψ)ω2).The formsωi, ϑi, and αi define the same structure on M and we let ω12, ϑ12, and α12 be the connection forms associated to the coframes ω1, ω2; ϑ1, ϑ2; α1, α2. The next Theorem is crucial for what follows.Theorem 8.
(44)
ϑ
12
=
d
ψ
+
ω
12
+
*
d
log
⁡
A
=
2
d
ψ
+
α
12
.Proof.
Each of the transformations which yield theϑi and αi in the form (43) can be thought of as a composition of the two transformations which occur in the Theorems 1 and 7. First apply the transformation ωi→Aωi and τ→-ψ with ωi*→ϑi in (16), we get the ϑi equations in (43). Invoking Theorems 1 and 7 in turn, the first result is obtained
(45)ϑ12=dψ+ω12+*dlog⁡A.
The transformation to the αi is exactly similar except that τ→ψ, hence
(46)α12=-dψ+ω12+*dlog⁡A.
This implies that *dlog⁡A=α12+dψ-ω12. When replaced in the first equation of (44), the second equation appears. Note that from Theorem 5, α12=α2, so the second equation can be given as ϑ12=2dψ+α2.Differentiating the second equation in (14) and using dα1=0, it follows that
(47)d*ω12=0.Lemma 9.
The angleψ is a harmonic function d*dψ=0 and moreover, d*ϑ12=0.Proof.
From Theorem8, it follows by applying * through (44) that
(48)*ϑ12=*ω12+*dψ-dlog⁡A=2*dψ-α1.
Exterior differentiation of this equation using d*ω12=0 immediately gives
(49)d*dψ=0.
This states that ψ is a harmonic function. Equation (48) also implies that d*ϑ12=0.


## 5. Construction of the Closed Differential Ideal Associated withM

Exterior differentiation of the first equation in (14) and using the second equation produces
(50)dϑ1+(α1+2*ω12)∧ϑ1=0.
The structure equation for the ϑi will be needed,
(51)dϑ1=ϑ12∧ϑ2=-*ϑ12∧ϑ1.
From the second equation in (44), we have *ω12-dlog⁡A+α1=*dψ, and putting this in the first equation of (44) we find
(52)-*ϑ12+α1+2*ω12=2dlog⁡A.
Using (52) in (51),
(53)dϑ1+(α1+2*ω12)∧ϑ1=2dlog⁡A∧ϑ1.
Replacing dϑ1 by means of (50) implies the following important result
(54)dlog⁡A∧ϑ1=0.
Equation (54) and Cartan’s lemma imply that there exists a function B such that
(55)dlog⁡A=Bϑ1.
This is the first in a series of results which relates many of the variables in question such as J, B, and ϑ12 directly to the one-form ϑ1. To show this requires considerable work. The way to proceed is to use the forms αi in Theorem 5 because their exterior derivatives are known. For an arbitrary function on M, define
(56)df=f1α1+f2α2.
Differentiating (56) and extracting the coefficient of α1∧α2, we obtain
(57)f21-f12+f2=0.
In terms of the αi, *dψ=ψ1α2-ψ2α1, Lemma 9 yields
(58)ψ11+ψ22+ψ1=0.
Finally, since *ϑ12=2*dψ-α1, substituting for *dψ, we obtain that
(59)*ϑ12=-(2ψ2+1)α1+2ψ1α2.
Differentiating structure equation (51) and using Lemma 9,
(60)*ϑ12∧dϑ1=0,
so,
(61)*ϑ12∧ϑ12∧ϑ2=0.
This equation implies that either ϑ12 or *ϑ12 is a multiple by a function of the form ϑ2. Hence, for some function p,
(62)ϑ12=-pϑ2,*ϑ12=pϑ1,ϑ12=pϑ1,*ϑ12=pϑ2.
Substituting the first line of (62) back into the structure equation, we have
(63)dϑ1=0.
The second line yields simply dϑ1=pϑ1∧ϑ2. Only the first case is examined now. Substituting (63) into (50), the following important constraint is obtained
(64)(α1+2*ω12)∧ϑ1=0.Theorem 10.
The functionψ satisfies the equation
(65)2ψ1cos⁡(2ψ)+(2ψ2+1)sin(2ψ)=0.Proof.
By substituting*dψ into (48) we have
(66)*ϑ12=2*(ψ1α1+ψ2α2)-α1=-(2ψ2+1)α1+2ψ1α2.
Substituting (66) into (44) and solving for *ω12, we obtain that
(67)*ω12=*ϑ12-*dψ+dlog⁡A=*ϑ12-*dψ+Bϑ1=*dψ-α1+Bϑ1.
This can be put in the equivalent form
(68)2*ω12+α1=2*dψ-α1+2Bϑ1.
Taking the exterior product with ϑ1 and using dψ1, we get
(69)(α1+2*ω12)∧ϑ1=(2*dψ-α1)∧ϑ1=(2ψ1*α1+2ψ2*α2-α1)∧ϑ1=(2ψ1cos⁡(2ψ)+(2ψ2+1)sin(2ψ))ϑ2∧ϑ1.
Imposing the constraint (64), the coefficient of ϑ1∧ϑ2 can be equated to zero. This produces the result (65).As a consequence of Theorem10, a new function C can be introduced such that
(70)2ψ1=Csin(2ψ),2ψ2+1=-Ccos⁡(2ψ).
Differentiating each of these with respect to the αi basis, we get for i=1,2 that
(71)2ψ1i=Cisin(2ψ)+2ψiCcos⁡(2ψ),2ψ2i=-Cicos⁡(2ψ)+2ψiCsin(2ψ).
Substituting f=ψ into (57) and using the fact that ψ satisfies (58) gives the pair of equations
(72)-C1cos⁡(2ψ)-C2sin(2ψ)+2ψ1Csin(2ψ)-(2ψ2+1)Ccos⁡(2ψ)-1=0,C1sin(2ψ)-C2cos⁡(2ψ)+2ψ1Ccos⁡(2ψ)+(2ψ2+1)Csin(2ψ)=0.
This linear system can be solved for C1 and C2 to get
(73)C1+C(2ψ2+1)+cos⁡(2ψ)=0,C2-2Cψ1+sin(2ψ)=0.
By differentiating each of the equations in (73), it is easy to verify that C satisfies (57), namely, C12-C21-C2=0. Hence there exist harmonic functions which satisfy (65). The solution depends on two arbitrary constants, the values of ψ and C at an initial point.Lemma 11.
Consider(74)dC=(C2-1)ϑ1,*ϑ12=Cϑ1.Proof.
It is easy to express theϑi in terms of the αi,
(75)ϑ1=cos⁡(2ψ)α1+sin(2ψ)α2,ϑ2=-sin(2ψ)α1+cos⁡(2ψ)α2.
Therefore, using (70) and (73), it is easy to see that
(76)dC=C1α1+C2α2=(C2-1)(cos⁡(2ψ)α1+sin(2ψ)α2)=(C2-1)ϑ1.
Using (70), it follows that
(77)*ϑ12=-(2ψ2+1)α1+2ψ1α2=Ccos⁡(2ψ)α1+Csin(2ψ)α2=C(cos⁡(2ψ)α1+sin(2ψ)α2)=Cϑ1.
This implies that ϑ12=-Cϑ2.It is possible to obtain formulas forB1, B2. Using (75) in (55), the derivatives of log⁡A can be written down
(78)(log⁡A)1=Bcos⁡(2ψ),(log⁡A)2=Bsin(2ψ).
Differentiating each of these in turn, we obtain for i=1,2,
(79)(log⁡A)1i=Bicos⁡(2ψ)-2Bψisin(2ψ),(log⁡A)2i=Bisin(2ψ)+2Bψicos⁡(2ψ).
Taking f=log⁡A in (57) produces a first equation for the Bi,
(80)B1sin(2ψ)+2Bψ1cos⁡(2ψ)-B2cos⁡(2ψ)+2Bψ2sin(2ψ)+Bsin(2ψ)=0.
If another equation in terms of B1 and B2 can be found, it can be solved simultaneously with (80). There exists such an equation and it can be obtained from the Gauss equation in (4) which we put in the form
(81)dω12=-acω1∧ω2=-acA-2α1∧α2.Solving (44) for ω12, we have
(82)ω12=dψ+α2+(log⁡A)2α1-(log⁡A)1α2.
The exterior derivative of this takes the form,
(83)dω12=[1-(log⁡A)11-(log⁡A)22-(log⁡A)1]α1∧α2.
Putting this in the Gauss equation,
(84)-(log⁡A)11-(log⁡A)22+{-(log⁡A)1+1}+acA-2=0.
Replacing the second derivatives from (79), we have the required second equation
(85)-B1cos⁡⁡(2ψ)-B2sin⁡(2ψ)+B{2ψ1sin(2ψ)-(2ψ2+1)cos⁡(2ψ)}+1+acA-2=0.
Solving (80) and (85) together, the following expressions for B1 and B2 are obtained
(86)B1+B(2ψ2+1)-(1+acA-2)cos⁡(2ψ)=0,B2-2Bψ1-(1+acA-2)sin(2ψ)=0.
Given these results for B1 and B2, it is easy to produce the following two Lemmas.Lemma 12.
Consider(87)dB=(BC+1+acA-2)ϑ1,dlog⁡J=(C+2B)ϑ1.Proof.
Substituting (86) into dB, we get
(88)dB=B1α1+B2α2=(BC+1+acA-2)(cos⁡⁡(2ψ)α1+sin⁡(2ψ)α2)=(BC+1+acA-2)ϑ1.
Moreover,
(89)dlog⁡J=α1+2*ω12=α1+2(*ϑ12-*dψ+dlog⁡A)=α1+2*ϑ12-2*dψ+2dlog⁡A=*ϑ12+2dlog⁡A=Cϑ1+2Bϑ1.Lemma 13.
Consider(90)dψ=-12sin(2ψ)ϑ1-12(C+cos⁡(2ψ))ϑ2.Proof.
Consider(91)2dψ=2ψ1α1+2ψ2α2=Csin(2ψ)α1-(Ccos⁡(2ψ)+1)α2=Csin(2ψ)(cos⁡(2ψ)ϑ1-sin(2ψ)ϑ2)-(Ccos⁡(2ψ)+1)(sin(2ψ)ϑ1+cos⁡(2ψ)ϑ2)=-sin(2ψ)ϑ1-(C+cos⁡(2ψ))ϑ2.In the interests of completeness, it is important to verify the following Theorem.Theorem 14.
The functionB satisfies (57) provided ψ satisfies both (58) and (64).Proof.
DifferentiatingB1 and B2 given by (86), the left side of (57) is found to be
(92)B21-B12+B2=2B1ψ1+B2(2ψ2+1)+2B(ψ11+ψ22+ψ1)+A-2((ac)1sin⁡(2ψ)-(ac)2sin⁡(2ψ))-2acBA-2(cos⁡⁡(2ψ)sin⁡(2ψ)-sin⁡(2ψ)cos⁡⁡(2ψ))+(1+acA-2)(2ψ1cos⁡⁡(2ψ)+(2ψ2+1)sin⁡(2ψ))=2(1+acA-2)(2ψ1cos⁡⁡(2ψ)+(2ψ2+1)sin⁡(2ψ))+A-2((ac)1sin(2ψ)-(ac)2cos⁡(2ψ)).
To simplify this, (58) has been substituted. Using (75) and *d(ac)=(ac)1α2-(ac)2α1, it follows that
(93)*d(ac)∧ϑ2=((ac)1sin(2ψ)-(ac)2cos⁡(2ψ))α1∧α2.
Note that the coefficient of α1∧α2 in this appears in the compatibility condition. To express it in another way, begin by finding the exterior derivative of 4ac=(a+c)2-(a-c)2,
(94)4d(ac)=2(a+c)(a-c)ϑ1-2(a-c)2(α1+2*ω12).
Applying the Hodge operator to both sides of this gives, upon rearranging terms,
(95)2*d(ac)a-c=(a+c)ϑ2-(a-c)(α2-2ω12).
Consequently, we can write
(96)-2(a-c)2*d(ac)∧ϑ2=(α2-2ω12)∧ϑ2=-(2ψ1cos⁡(2ψ)+(2ψ2+1)sin(2ψ))α1∧α2.
Therefore, it must be that
(97)-(ac)1sin(2ψ)+(ac)2cos⁡(2ψ)=-12(a-c)2(2ψ1cos⁡(2ψ)+(2ψ2+1)sin(2ψ)).
It follows that when f=B, (57) finally reduces to the form
(98)(1+H2A-2)[2ψ1cos⁡(2ψ)+(2ψ2+1)sin(2ψ)]=0.
The first factor is clearly nonzero, so the second factor must vanish. This of course is equivalent to the constraint (64).


## 6. Intrinsic Characterization ofM

During the prolongation of the exterior differential system, the additional variablesψ, A, B, and C have been introduced. The significance of the appearance of the function C, is that the process terminates and the differentials of all these functions can be computed without the need to introduce more functions. This means that the exterior differential system has finally closed.The results of the previous section, in particular the lemmas, can be collected such that they justify the following.Proposition 15.
The differential system generated in terms of the differentials of the variablesψ, A, B, and C is closed. The variables H, J, A, B, and C remain constant along the ϑ2-curves so ϑ1=0. Hence, an isometry that preserves H must map the ϑ1, ϑ2 curves onto the corresponding ϑ1*, ϑ2* curves of the associated surface M* which is isometric to M.
Along theϑ1, ϑ2 curves, consider the normalized frame,
(99)ζ1=cos⁡(ψ)e1+sin(ψ)e2,ζ2=-sin(ψ)e1+cos⁡(ψ)e2.
The corresponding coframe and connection form are
(100)ξ1=cos⁡(ψ)ω1+sin(ψ)ω2,ξ2=-sin(ψ)ω1+cos⁡(ψ)ω2,ξ12=dψ+ω12.
Then ϑ1 can be expressed as a multiple of ξ1 and ϑ2, ϑ12 in terms of ξ2, and the differential system can be summarized here:
(101)ϑ1=Aξ1,ϑ2=Aξ2,ϑ12=ξ12+*dlog⁡A=-CAξ2,dlog⁡A=ABξ1,dB=A(BC+1+acA-2)ξ1,dC=A(C2-1)ξ1,dH=AJξ1,dJ=AJ(2B+C)ξ1.
The condition dϑ1=0 is equivalent to
(102)dA∧ξ1+Adξ1=0.
This implies that dξ1=0 since dA is proportional to ξ1. Also d*ϑ12=0 is equivalent to d*ξ12=0.
Moreover,d*ξ12=0 is equivalent to the fact that the ξ1, ξ2-curves can be regarded as coordinate curves parameterized by isothermal parameters. Therefore, along the ξ1, ξ2-curves, orthogonal isothermal coordinates denoted (s,t) can be introduced. The first fundamental form of M then takes the form,
(103)I=ξ12+ξ22=E(s)(ds2+dt2).
Now suppose we set e(s)=E(s), then
(104)ξ1=e(s)ds,ξ2=e(s)dt,ξ12=e′(s)e2(s)ξ2=e′(s)e(s)dt.
This means such a surface is isometric to a surface of revolution. Since ψ, d*ξ12=0, (100) implies that d*ω12=0. This can be stated otherwise as the principal coordinates are isothermal and so M is an isothermic surface.
SinceA, B, C, H, and J are functions of only the variable s, this implies that H and J or H and K are constant along the t-curves where s is constant. This leads to the following proposition.Proposition 16.
(105)
d
H
∧
d
K
=
0
,
ξ
12
=
-
(
C
+
B
)
A
ξ
2
.
This is equivalent to the statement thatM is a Weingarten surface.Proof.
The first result follows from the statement about the coordinate system above. Sinceϑ12=ξ12+*dlog⁡A=-CAξ2 and dA=A2Bξ1,
(106)ξ12=-CAξ2-*dlog⁡A=-CAξ2-*A-1dA=-CAξ2-AB*ξ1=-(C+B)Aξ2.
Consequently, the geodesic curvature of each ξ2-curve, s constant is
(107)e′(s)e2(s)=-A(B+C),
which is constant.To express theωi in terms of ds and dt, start by writing ωi in terms of the ξi and then substituting (104),
(108)ω1=cos⁡(ψ)eds-sin(ψ)edt,ω2=sin(ψ)eds+cos⁡(ψ)edt.
Subscripts (s,t) denote differentiation and Hs=H′ is used interchangeably. Beginning with dH=H′ds and using (108), we have
(109)dH=H1ω1+H2ω2=(H1cos⁡(ψ)+H2sin(ψ))eds+(-H1sin(ψ)+H2cos⁡(ψ))edt=H′ds.
Equating coefficients of differentials, this implies that
(110)H1ecos⁡(ψ)+H2esin(ψ)=H′,-H1sin(ψ)+H2cos⁡(ψ)=0.
Solving this as a linear system we obtain H1, H2,
(111)H1=H′ecos⁡(ψ),H2=H′esin(ψ).
Noting that u=H1/J and v=H2/J, using (100) the forms αi can be expressed in terms of ds, dt:
(112)α1=H′J(cos⁡(2ψ)ds-sin(2ψ)dt),α2=H′J(sin(2ψ)ds+cos⁡(2ψ)dt).
Substituting ξ1 from (104) into dH=AJξ1,
(113)dH=H′ds=AJξ1=AJe(s)ds.
Therefore, H′=AJe>0 and so H(s) is an increasing function of s. Now define the function Q(s) to be
(114)Q=H′J=A·e>0.
Substituting (114) into (112), the αi are expressed in terms of Q as well. The equations (30) in Theorem 5 can easily be expressed in terms of ψ and Q.Theorem 17.
Equations (30) are equivalent to the following system of coupled equations in ψ and Q:
(115)sin(2ψ)(log⁡(Q))s+2cos⁡(2ψ)ψs-2sin(2ψ)ψt=0,cos⁡(2ψ)(log⁡(Q))s-2sin(2ψ)ψs-2cos⁡(2ψ)ψt=Q.
Moreover, (115) are equivalent to the following first-order system:
(116)ψs=-12Qsin(2ψ),ψt=12(log⁡(Q))s-12Qcos⁡(2ψ).
System (116) can be thought of as a type of Lax pair. Moreover, (116) implies that ψ is harmonic as well. Differentiating ψs with respect to s and ψt with respect to t, it is clear that ψ satisfies Laplace’s equation in the (s,t) variables ψss+ψtt=0. This is another proof that ψ is harmonic.Theorem 18.
The functionQ(s) satisfies the following second-order nonlinear differential equation:
(117)Q′′(s)Q(s)-(Q′(s))2=Q4(s).
There exists a first integral for this equation of the following form:
(118)Q′(s)2=Q(s)4+κQ(s)2,κ∈R.Proof.
Equation (117) is just the compatibility condition for the first-order system (116). The required derivatives are
(119)ψst=-Q2cos⁡(2ψ)((log⁡Q)s-Qcos⁡(2ψ)),ψts=12(log⁡Q)ss-12Qscos⁡(2ψ)+Qsin(2ψ)ψs.
Equating derivatives ψst=ψts, the required (117) follows.
Differentiating both sides of (118) we get
(120)Q′′(s)=2Q(s)3+κQ(s).
Isolating κQ(s) from (118) and substituting it into (120), (117) appears.It is important to note that the functionC which appears when the differential ideal closes can be related to the function Q.Corollary 19.
(121)
C
=
(
1
Q
)
′
.Proof.
Usingϑi from (101) in Lemma 13, in the s, t coordinates
(122)2dψ=-sin⁡(2ψ)Aeds-(C+cos⁡(2ψ))Aedt=ψsds+ψtdt.
Hence using (116), this implies that 2ψs=-sin(2ψ)Ae=-Qsin(2ψ), hence Q=Ae. The second equation in (116) for ψt implies that (C+cos⁡(2ψ))Ae=Qcos⁡(2ψ)-(log⁡Q)′. Replacing Ae=Q, this equation simplifies to the form (121).


## 7. Integrating the Lax Pair System

It is clear that the first order equation in (116) for Q(s) is separable and can be integrated. The integral depends on whether K is zero or nonzero:
(123)Q(s)=1ϵs+γ,K=0;log⁡(2(K+KQ2+K)Q)=ϵKs+γ,K≠0.
Here, ϵ=±1 and γ is the last constant of integration. Taking specific choices for the constants, for example, eγ=2K when K≠0 and a=K, the set of solutions (123) for Q(s) can be summarized below:
(124)Dom⁡(s)Q(s)Dom⁡(s)Q(s)s>01ss<0-1s0<s<πaasin(as)-πa<s<0-asin(as)s>0asinh⁡(as)s<0-asinh⁡⁡(as).
It is presumed that other choices of the constants can be geometrically eliminated in favor of (124). The solutions (124) are then substituted back into linear system (116). The first equation in (116) implies that either of the two cases
(125)ψ≡0,mod⁡π2;2ψssin(2ψ)=-Q.
Substitute ψ≡0 into the second equation in (116). It implies that (log⁡Q)s=Q and ψ=π/2 give (log⁡Q)s=-Q. In both cases Q(s) is a solution which already appears in (124).For the second case in (125), the equation can be put in the form
(126)(log⁡|tan(ψ)|)s=-Q.
Integrating we have for some function y(t) to be determined,
(127)tan(ψ)=e-∫‍Q(s)ds·y(t).
Therefore, tan(ψ) can be obtained by substituting for Q(s) for each of the three cases in (124). The upper sign holds for s>0 and the lower sign holds if s<0.(i)
Q(s)=±s-1, -∫‍Q(s)ds=log⁡|s|∓, and
(128)tan(ψ)=s∓·y(t).(ii)
Q(s)=±(a/sin(as)), -∫‍Q(s)ds=log⁡|csc(as)-cot(as)|∓, and
(129)tan(ψ)=(tan(as2))∓·y(t).(iii)
Q(s)=±(a/sinh⁡(as)), -∫‍Q(s)ds=∓arctanh(eas), and
(130)tan(ψ)=(tanh(as2))∓·y(t).In case (ii), if s>0 and y(t)=±1 then ψ=±(1/2)(as+π), mod⁡π, and if s<0 and y(t)=±1, then ψ=±(1/2)as, mod⁡π.It remains to integrate the second equation of the Lax pair (116) using solutions for both Q(s) and tan(ψ). The first case (i) is not hard and will be shown explicitly here. The others can be done, and more complicated cases are considered in the Appendix.(i) ConsiderQ(s)=s-1 and tan(ψ)=s-1·y(t). The second equation in (116) simplifies considerably to yt=-1; therefore,
(131)y(t)=-(t+σ),tan(ψ)=-(t+σ)s.ForQ(s)=-s-1 and tan(ψ)=s·y(t), the second equation of (116) becomes yt=-y2; therefore,
(132)y(t)=1t+σ,tan(ψ)=st+σ.


## 8. A Third Order Equation forH and Fundamental Forms

Sinceξ12=(log⁡e(s))′dt, using (104) ω12 can be written as
(133)ω12=ξ12-dψ=(log⁡e(s))′dt-dψ.
Using (14) and (112) for α1, it follows that
(134)dlog⁡(J)=Q(cos⁡(2ψ)ds-sin(2ψ)dt)-2*(ψtdt+ψsds)+2*(log⁡(e(s)))′dt.
When the ωi are put in the s, t coordinates, using *ω1=ω2, it can be stated that *ds=dt and *dt=-ds. Consequently, dlog⁡(J) simplifies to
(135)dlog⁡(J)=(Qcos⁡(2ψ)+2ψt-2(log⁡(e(s)))′)ds+(-Qsin(2ψ)-2ψs)dt.
First order system (116) permits this to be written using e(s)=E(s) as
(136)(log⁡(J))′+(log⁡(E))′=(log⁡(Q))′.
Hence there exists a constant τ independent of s such that E·J=τQ or
(137)E=τQJ=τQ2H′.
This result (137) for E is substituted into the Gauss equation -((log⁡(E))ss+(log⁡(E))tt)=2E(H2-J2) giving
(138)(log⁡(E))′′=2(log⁡(Q))′′-(log⁡(Hs))′′=2Q2-(H′′H′)′.
Therefore, the Gauss equation transforms into a third-order differential equation in the s variable,
(139)(H′′H′)′+2τH=2Q2(1+τH2H′).
Thus a characterization of Bonnet surfaces is reached by means of the solutions to these equations. This equation determines the function H(s) and after that the functions J(s) and E(s). Therefore, Bonnet surfaces have as first fundamental form the expression
(140)I=E(s)(ds2+dt2),E(s)=τQ2(s)H′(s).
Since ψ is the angle from the principal axis e1 to the s-curve with t equals constant, the second fundamental form is given by
(141)II=Lds2+2Mdsdt+Ndt2,
where the coefficients L, M, and N are given by
(142)L=E(H+Jcos⁡(2ψ))=EH+τQcos⁡(2ψ),M=-EJsin(2ψ)=-τQsin(2ψ),N=E(H-Jcos⁡(2ψ)).



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*Source: 715679-2014-08-31.xml*