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Mathematics » "Manifolds - Current Research Areas", book edited by Paul Bracken, ISBN 978-953-51-2872-4, Print ISBN 978-953-51-2871-7, Published: January 18, 2017 under CC BY 3.0 license. © The Author(s).

Chapter 2

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

By Paul Bracken
DOI: 10.5772/67008

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

Paul Bracken
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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.

Keywords: manifold, differential form, closed, isometric, differential equation, Bonnet surface

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 admits 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 [35]. Here the approach first given by Chern [6] to Bonnet surfaces is considered. The development 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 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 surface M in E3 with nonconstant mean curvature function H. Moreover, suppose M admits infinitely many nontrivial and geometrically distinct isometries preserving H. Suppose U is the set of umbilic points of M and V 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, UV [7, 8]. Interestingly, there is no point in VU 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 [911]. 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

Over M, there exists a well‐defined field of orthonormal frames, which is written as x, e1,e2,e3 such that xM, e3 is the unit normal at x, and e1,e2 are along principal directions [12]. The fundamental equations for M have the form

dx=ω1e1+ω2e2,de1=ω12e2+ω13e3,de2=ω12e1+ω23e3,de3=ω13e1ω23e2.
(1)

Differentiating each of these equations in turn, results in a large system of equations for the exterior derivatives of the ωi and 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

ω12=hω1+kω2,ω13=aω1,ω23=cω2.
(2)

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

H=12(a+c),K=ac.
(3)

The forms which appear in Eq. (1) satisfy the fundamental structure equations which are summarized here [14],

dω1=ω12ω2,dω2=ω1ω12dω13=ω12ω23dω23=ω13ω12,dω12=acω2ω1=Kω1ω2.
(4)

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

(da(ac)hω2)ω1=0,(dc(ac)kω1)ω2=0.
(5)

Cartan's lemma can be applied to the equations in (5). Thus, there exist two functions u and v such that

1acdahω2=(uk)ω1,1acdckω1=(vh)ω2.
(6)

Subtracting the pair of equations in (6) gives an expression for dlog(ac)

dlog(ac)=(u2k)ω1(v2h)ω2.
(7)

Define the variable J to be

It will appear frequently in what follows. Equation (7) then takes the form

dlogJ=(u2k)ω1(v2h)ω2.
(9)

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,

ϑ1=uω1+vω2,ϑ2=vω1+uω2,α1=uω1vω2,α2=vω1+uω2.
(10)

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 relation 2H=a+c and subtracting the relation 4K=4ac yields 4(H2K)=(ac)2. The Hodge operator, denoted by *, will play an important role throughout. It produces the following result on the basis forms ωi,

*ω1=ω2,*ω2=ω1,*2=1.
(11)

Moreover, adding the expressions for da and dc given in Eq. (6), there results

1ac(da+dc)=(uk)ω1+hω2+(vk)ω2+kω1=uω1+vω2=ϑ1.
(12)

Finally, note that

α1+2*ω12=uω1vω2+2*(hω1+kω2)=(u2k)ω1(v2h)ω2=dlogJ.
(13)

Therefore, the Codazzi equations (12) and (13) can be summarized using the definitions of H and J as

dH=Jϑ1,dlogJ=α1+2*ω12.
(14)

3. A theorem of Bonnet

Suppose that M* is a surface which is isometric to M such that the principal curvatures are preserved [1012]. Denote all quantities which pertain to M* with the same symbols but with asterisks, as for example

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

ω1*=cosτω1sinτω2,ω2*=sinτω1+cosτω2.
(15)

Theorem 3.1 Under the transformation of coframe given by Eq. (15), the associated connection forms are related by

ω12*=ω12dτ.
(16)

Proof: Exterior differentiation of ω1* produces

dω1*=sinτdτω1+cosτdω1cosτdτω2sinτdω2=dτ(sinτω1cosτω2)+cosτω12ω2sinτω1ω12=(dτ+ω12)ω2*.

Similarly, differentiating ω2* gives

dω2*=cosτω1+sinτdω1sinτdτω2+cosτdω2=dτ(cosτω1sinτω2)+sinτω12ω2+cosτω1ω12=ω1*(dτ+ω12).

There is a very important result that can be developed at this point. In the case that a=a* and c=c*, the Codazzi equations imply that

α1+2*ω12=dlog(ac)=dlog(a*c*)=α1*+2*ω12*.

Apply the operator * to both sides of this equation, we obtain

α22ω12=α2*2ω12*.

Substituting for ω12* from Theorem 3.1, this is

2dτ=α2α2*.
(17)

Lemma 3.1

ϑ1=ϑ1*.

Proof: This can be shown in two ways. First from Eq. (15), express the ωi in terms of the ωi*

ω1=cosτω1*+sinτω2*,ω2=sinτω1*+cosτω2*.
(18)

Therefore,

ϑ1=uω1+vω2=u(cosτω1*+sinτω2*)+v(sinτω1*+cosτω2*)=u*ω1*+v*ω2*=ϑ1*,

where u*=ucosτvsinτ and v*=usinτ+vcosτ.

Lemma 3.1 also follows from the fact that dH=dH* and Eq. (8).

Lemma 3.2

α2*=sin(2τ)α1+cos(2τ)α2.

Proof:

α2*=(usinτ+vcosτ)(cosτω1sinτω2)+(ucosτvsinτ)(sinτω1+cosτω2) =(usin(2τ)+vcos(2τ))ω1+(vsin(2τ)+ucos(2τ))ω2 =sin(2τ)α1+cos(2τ)α2.

Substituting α2* from Lemma 3.2 into Eq. (13), dτ can be written as

dτ=12(α2sin(2τ)α1cos(2τ)α2)=12((1cos(2τ))α2sin(2τ)α1).
(19)

Introduce the new variable t=cot(τ) so dt=csc2(τ)dτ and sinτ=11+t2, cosτ=11+t2, hence the following lemma.

Lemma 3.3

dt=tα1α2.

This is the total differential equation which must be satisfied by the angle τ 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 3.2 A surface M admits a nontrivial isometric deformation that keeps the principal curvatures fixed if and only if

dα1=0,dα2=α1α2.
(20)

Proof: Differentiating both sides of Lemma 3.3 gives

dtα1+tdα1dα2=(tα1α2)α1+tdα1dα2=0.

Equating the coefficients of t to zero gives the result (20).

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 Eq. (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 is linearly independent, t equals a constant. Thus, we arrive at a theorem originally due to Bonnet.

Theorem 3.3 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 forms ω1,ω2, the first two of the structure equations uniquely determine the form ω12. The ω1,ω2 is called the orthonormal coframe of the metric

ds2=ω12+ω22,

and ω12 is the connection form associated with it.

Theorem 4.1 Suppose that A>0 is a function on M. Under the change of coframe

ω1*=Aω1,ω2*=Aω2,
(21)

the associated connection forms are related by

ω12*=ω12+*dlogA.
(22)

Proof: The structure equations for the transformed system are given as

dω1*=ω12*ω2*,dω2*=ω1*ω12*.

Using Eq. (21) to replace the ωi* in these, we obtain

dlogAω1+dω1=ω12*ω2,dlogAω2+dω2=ω1ω12*.

The ωi satisfy a similar system of structure equations, so replacing dωi here yields

(ω12*ω12)ω2=dlogAω1,(ω12*ω12)ω1=dlogAω2.

Since the form ωi satisfies 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

ω1*(ω12*ω12)=ω1dlogA,ω2*(ω12*ω12)=ω2dlogA.

Cartan's lemma can be used to conclude from these that there exist functions f and g such that

*(ω12*ω12)=dlogAfω1,*(ω12*ω12)=dlogA+gω2.

Finally, apply * to both sides and use *2=1 to obtain

ω12*ω12=*dlogA+fω2,ω12*ω12=*dlogA+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, Theorem 3.2, 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, MV 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

To define more general transformations of the ωi, define the angle ψ as

u=Acos(ψ),v=Asin(ψ).
(24)

This angle, which is defined modulo 2π, is continuous only locally and could be discontinuous in a nonsimply connected region of MV. With A and ψ related to u and v by Eq. (24), the forms ϑi and αi can be written in terms of A and ψ as

ϑ1=A(cos(ψ)ω1+sin(ψ)ω2),ϑ2=A(sin(ψ)ω1+cos(ψ)ω2),α1=A(cos(ψ)ω1sin(ψ)ω2),α2=A(sin(ψ)ω1+cos(ψ)ω2).
(25)

The forms ωi, ϑi, αi define the same structure on M and we let ω12, ϑ12, α12 be the connection forms associated to the coframes ω1,ω2; ϑ1,ϑ2; α1,α2. The next theorem is crucial for what follows.

Theorem 4.2

ϑ12=dψ+ω12+*dlogA=2dψ+α12.
(26)

Proof: Each of the transformations which yield the ϑi and αi in the form (25) can be thought of as a composition of the two transformations which occur in the Theorems 3.1 and 4.1. First apply the transformation ωiAωi and τψ with ωi*ϑi in Eq. (15), we get the ϑi equations in Eq. (25). Invoking Theorems 3.1 and 4.1 in turn, the first result is obtained

ϑ12=dψ+ω12+*dlogA.

The transformation to the αi is exactly similar except that τψ, hence

α12=dψ+ω12+*dlogA.

This implies *dlogA=α12+dψω12. When replaced in the first equation of (26), the second equation appears. Note that from Theorem 3.2, α12=α2, so the second equation can be given as ϑ12=2dψ+α2.

Differentiating the second equation in Eq. (14) and using dα1=0, it follows that

Lemma 4.1 The angle ψ is a harmonic function d*dψ=0 and moreover, d*ϑ12=0.

Proof: From Theorem 4.2, it follows by applying * through Eq. (26) that

*ϑ12=*ω12+*dψdlogA=2*dψα1.
(28)

Exterior differentiation of this equation using d*ω12=0 immediately gives

d*dψ=0.

This states that ψ is a harmonic function. Equation (28) also implies that d*ϑ12=0.

5. Construction of the closed differential ideal associated with M

Exterior differentiation of the first equation in (14) and using the second equation produces

dϑ1+(α1+2*ω12)ϑ1=0.
(29)

The structure equation for the ϑi will be needed,

dϑ1=ϑ12ϑ2=*ϑ12ϑ1.
(30)

From the second equation in Eq. (26), we have *ω12dlogA+α1=*dψ, and putting this in the first equation of Eq. (26), we find

*ϑ12+α1+2*ω12=2dlogA.
(31)

Using Eq. (31) in Eq. (30),

dϑ1+(α1+2*ω12)ϑ1=2dlogAϑ1.
(32)

Replacing dϑ1 by means of Eq. (29) implies the following important result

Equation (33) and Cartan's lemma imply that there exists a function B such that

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 3.2 because their exterior derivatives are known. For an arbitrary function on M, define

df=f1α1+f2α2.
(35)

Differentiating Eq. (35) and extracting the coefficient of α1α2, we obtain

f21f12+f2=0.
(36)

In terms of the αi, *dψ=ψ1α2ψ2α1, Lemma 4.1 yields

ψ11+ψ22+ψ1=0.
(37)

Finally, since *ϑ12=2*dψα1, substituting for *dψ, we obtain that

*ϑ12=(2ψ2+1)α1+2ψ1α2.
(38)

Differentiating structure equation (30) and using Lemma 4.1,

*ϑ12dϑ1=0,

so,

*ϑ12ϑ12ϑ2=0

This equation implies that either ϑ12 or *ϑ12 is a multiple by a function of the form ϑ2. Hence, for some function p,

ϑ12=pϑ2,*ϑ12=pϑ1,ϑ12=pϑ1,*ϑ12=pϑ2,
(39)

Substituting the first line of Eq. (39) back into the structure equation, we have

The second line yields simply dϑ1=pϑ1ϑ2. Only the first case is examined now. Substituting Eq. (40) into Eq. (29), the following important constraint is obtained

(α1+2*ω12)ϑ1=0.
(41)

Theorem 5.1 The function ψ satisfies the equation

2ψ1cos(2ψ)+(2ψ2+1)sin(2ψ)=0.
(42)

Proof: By substituting *dψ into Eq. (28) we have

*ϑ12=2*(ψ1α1+ψ2α2)α1=(2ψ2+1)α1+2ψ1α2.
(43)

Substituting Eq. (43) into Eq. (26) and solving for *ω12, we obtain that

*ω12=*ϑ12*dψ+dlogA=*ϑ12*dψ+Bϑ1=*dψα1+Bϑ1.

This can be put in the equivalent form

2*ω12+α1=2*dψα1+2Bϑ1.
(44)

Taking the exterior product with ϑ1 and using dψ1, we get

(α1+2*ω12)ϑ1=(2*dψα1)ϑ1=(2ψ1*α1+2ψ2*α2α1)ϑ1=(2ψ1cos(2ψ)+(2ψ2+1)sin(2ψ))ϑ2ϑ1.

Imposing the constraint (41), the coefficient of ϑ1ϑ2 can be equated to zero. This produces the result (42).

As a consequence of Theorem 5.1, a new function C can be introduced such that

2ψ1=Csin(2ψ),2ψ2+1=Ccos(2ψ).
(45)

Differentiation of each of these with respect to the αi basis, we get for i=1,2 that

2ψ1i=Cisin(2ψ)+2ψiCcos(2ψ),2ψ2i=Cicos(2ψ)+2ψiCsin(2ψ).

Substituting f=ψ into Eq. (36) and using the fact that ψ satisfies Eq. (37) gives the pair of equations

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

C1+C(2ψ2+1)+cos(2ψ)=0,C22Cψ1+sin(2ψ)=0.
(46)

By differentiating each of the equations in (46), it is easy to verify that C satisfies Eq. (36), namely, C12C21C2=0. Hence, there exist harmonic functions which satisfy Eq. (42). The solution depends on two arbitrary constants, the values of ψ and C at an initial point.

Lemma 5.1

dC=(C21)ϑ1,*ϑ12=Cϑ1.
(47)

Proof: It is easy to express the ϑi in terms of the αi,

ϑ1=cos(2ψ)α1+sin(2ψ)α2,ϑ2=sin(2ψ)α1+cos(2ψ)α2.
(48)

Therefore, using Eqs. (45) and (46), it is easy to see that

dC=C1α1+C2α2=(C21)(cos(2ψ)α1+sin(2ψ)α2)=(C21)ϑ1.

Using Eq. (45), it follows that

*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 for B1,B2. Using Eq. (48) in Eq. (34), the derivatives of logA can be written down

(logA)1=Bcos(2ψ),(logA)2=Bsin(2ψ).
(49)

Differentiating each of these in turn, we obtain for i=1,2,

(logA)1i=Bicos(2ψ)2Bψisin(2ψ),(logA)2i=Bisin(2ψ)+2Bψicos(2ψ).
(50)

Taking f=logA in Eq. (36) produces a first equation for the Bi,

B1sin(2ψ)+2Bψ1cos(2ψ)B2cos(2ψ)+2Bψ2sin(2ψ)+Bsin(2ψ)=0.
(51)

If another equation in terms of B1 and B2 can be found, it can be solved simultaneously with Eq. (51). There exists such an equation and it can be obtained from the Gauss equation in (4) which we put in the form

dω12=acω1ω2=acA2α1α2.

Solving Eq. (26) for ω12, we have

ω12=dψ+α2+(logA)2α1(logA)1α2.

The exterior derivative of this takes the form,

dω12=[1(logA)11(logA)22(logA)1]α1α2.

Putting this in the Gauss equation,

(logA)11(logA)22+{(logA)1+1}+acA2=0.

Replacing the second derivatives from Eq. (50), we have the required second equation

B1cos(2ψ)B2sin(2ψ)+B{2ψ1sin(2ψ)(2ψ2+1)cos(2ψ)}+1+acA2=0.
(52)

Solving Eqs. (51) and (52) together, the following expressions for B1 and B2 are obtained

B1+B(2ψ2+1)(1+acA2)cos(2ψ)=0,B22Bψ1(1+acA2)sin(2ψ)=0.
(53)

Given these results for B1 and B2, it is easy to produce the following two Lemmas.

Lemma 5.2

dB=(BC+1+acA2)ϑ1,dlogJ=(C+2B)ϑ1.
(54)

Proof: Substituting Eq. (53) into dB, we get

dB=B1α1+B2α2=(BC+1+acA2)(cos(2ψ)α1+sin(2ψ)α2)=(BC+1+acA2)ϑ1.

Moreover,

dlogJ=α1+2*ω12=α1+2(*ϑ12*dψ+dlogA)=α1+2*ϑ122*dψ+2dlogA=*ϑ12+2dlogA=Cϑ1+2Bϑ1.

Lemma 5.3

dψ=12sin(2ψ)ϑ112(C+cos(2ψ))ϑ2.
(55)

Proof:

2dψ=2ψ1α1+2ψ2α2=Csin(2ψ)α1(Ccos(2ψ)+1)α2=Csin(2ψ)(cos(2ψ)ϑ1sin(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 5.2 The function B satisfies Eq. (36) provided ψ satisfies both Eqs. (37) and (41).

Proof: Differentiating B1 and B2 given by Eq. (53), the left side of Eq. (36) is found to be

B21B12+B2=2B1ψ1+B2(2ψ2+1)+2B(ψ11+ψ22+ψ1)+A2((ac)1sin(2ψ)(ac)2sin(2ψ))2acBA2(cos(2ψ)sin(2ψ)sin(2ψ)cos(2ψ))+(1+acA2)(2ψ1cos(2ψ)+(2ψ2+1)sin(2ψ))=2(1+acA2)(2ψ1cos(2ψ)+(2ψ2+1)sin(2ψ))+A2((ac)1sin(2ψ)(ac)2cos(2ψ)).

To simplify this, Eq. (37) has been substituted. Using Eq. (48) and *d(ac)=(ac)1α2(ac)2α1, it follows that

*d(ac)ϑ2=((ac)1sin(2ψ)(ac)2cos(2ψ))α1α2.

Note that the coefficient of α1α2 in this appears in the compatibility condition. To express it in another way, begin by finding the exterior derivative of 4ac=(a+c)2(ac)2,

4d(ac)=2(a+c)(ac)ϑ12(ac)2(α1+2*ω12).

Applying the Hodge operator to both sides of this, gives upon rearranging terms

2*d(ac)ac=(a+c)ϑ2(ac)(α22ω12).

Consequently, we can write

2(ac)2*d(ac)ϑ2=(α22ω12)ϑ2=(2ψ1cos(2ψ)+(2ψ2+1)sin(2ψ))α1α2.

Therefore, it must be that

(ac)1sin(2ψ)+(ac)2cos(2ψ)=12(ac)2(2ψ1cos(2ψ)+(2ψ2+1)sin(2ψ)).

It follows that when f=B, Eq. (36) finally reduces to the form

(1+H2A2)[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 (41).

6. Intrinsic characterization of M

During the prolongation of the exterior differential system, the additional variables ψ, 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 6.1 The differential system generated in terms of the differentials of the variables ψ, A, B, and C is closed. The variables H,J,A,B,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 the ϑ1, ϑ2 curves, consider the normalized frame,

ζ1=cos(ψ)e1+sin(ψ)e2,ζ2=sin(ψ)e1+cos(ψ)e2.
(56)

The corresponding coframe and connection form are

ξ1=cos(ψ)ω1+sin(ψ)ω2,ξ2=sin(ψ)ω1+cos(ψ)ω2,ξ12=dψ+ω12.
(57)

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:

ϑ1=Aξ1,ϑ2=Aξ2,ϑ12=ξ12+*dlogA=CAξ2,dlogA=ABξ1,dB=A(BC+1+acA2)ξ1,dC=A(C21)ξ1,dH=AJξ1,dJ=AJ(2B+C)ξ1.
(58)

The condition dϑ1=0 is equivalent to

dAξ1+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,

I=ξ12+ξ22=E(s)(ds2+dt2).
(59)

Now suppose we set e(s)=E(s), then

ξ1=e(s)ds,ξ2=e(s)dt,ξ12=e(s)e2(s)ξ2=e(s)e(s)dt.
(60)

This means such a surface is isometric to a surface of revolution. Since ψ, d*ξ12=0, Eq. (57) implies that d*ω12=0. This can be stated otherwise as the principal coordinates are isothermal and so M is an isothermic surface.

Since A,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 6.2

dHdK=0,ξ12=(C+B)Aξ2.
(61)

This is equivalent to the statement M is a Weingarten surface.

Proof: The first result follows from the statement about the coordinate system above. Since ϑ12=ξ12+*dlogA=CAξ2 and dA=A2Bξ1,

ξ12=CAξ2*dlogA=CAξ2*A1dA=CAξ2AB*ξ1=(C+B)Aξ2

Consequently, the geodesic curvature of each ξ2 curve, s constant, is

e(s)e2(s)=A(B+C),

which is constant.

To express the ωi in terms of ds and dt, start by writing ωi in terms of the ξi and then substituting Eq. (60),

ω1=cos(ψ)edssin(ψ)edt,ω2=sin(ψ)eds+cos(ψ)edt.
(62)

Subscripts (s,t) denote differentiation and Hs=H is used interchangeably. Beginning with dH=Hds and using Eq. (62), we have

dH=H1ω1+H2ω2=(H1cos(ψ)+H2sin(ψ))eds+(H1sin(ψ)+H2cos(ψ))edt=Hds.

Equating coefficients of differentials, this implies that

H1ecos(ψ)+H2esin(ψ)=H,H1sin(ψ)+H2cos(ψ)=0.

Solving this as a linear system we obtain H1, H2,

H1=Hecos(ψ),H2=Hesin(ψ).
(63)

Noting that u=H1/J and v=H2/J, using Eq. (57) the forms αi can be expressed in terms of ds,dt

α1=HJ(cos(2ψ)dssin(2ψ)dt),α2=HJ(sin(2ψ)ds+cos(2ψ)dt).
(64)

Substituting ξ1 from Eq. (60) into dH=AJξ1,

dH=Hds=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

Q=HJ=Ae>0.
(65)

Substituting Eq. (65) into Eq. (64), αi is expressed in terms of Q as well. Equations (20) in Theorem 3.2 can easily be expressed in terms of ψ and Q.

Theorem 6.1 Equation (20) is equivalent to the following system of coupled equations in ψ and Q:

sin(2ψ)(log(Q))s+2cos(2ψ)ψs2sin(2ψ)ψt=0,cos(2ψ)(log(Q))s2sin(2ψ)ψs2cos(2ψ)ψt=Q.
(66)

Moreover, Eq. (66) is equivalent to the following first‐order system

ψs=12Qsin(2ψ),ψt=12(log(Q))s12Qcos(2ψ).
(67)

System (67) can be thought of as a type of Lax pair. Moreover, Eq. (67) 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 6.2 The function Q(s) satisfies the following second‐order nonlinear differential equation

Q(s)Q(s)(Q(s))2=Q4(s).
(68)

There exists a first integral for this equation of the following form

Q(s)2=Q(s)4+κQ(s)2,κR.
(69)

Proof: Equation (68) is just the compatibility condition for the first‐order system (67). The required derivatives are

ψst=Q2cos(2ψ)((logQ)sQcos(2ψ)),ψts=12(logQ)ss12Qscos(2ψ)+Qsin(2ψ)ψs.

Equating derivatives ψst=ψts, the required (68) follows.

Differentiating both sides of Eq. (69) we get

Q(s)=2Q(s)3+κQ(s).
(70)

Isolating κQ(s) from Eq. (69) and substituting it into Eq. (70), Eq. (68) appears.

It is important to note that the function C which appears when the differential ideal closes can be related to the function Q.

Corollary 6.1

Proof: Using ϑi from Eq. (58) in Lemma 5.3, in the s,t coordinates

2dψ=sin(2ψ)Aeds(C+cos(2ψ))Aedt=ψsds+ψtdt

Hence using Eq. (67), this implies that 2ψs=sin(2ψ)Ae=Qsin(2ψ), hence Q=Ae. The second equation in Eq. (67) for ψt implies that (C+cos(2ψ))Ae=Qcos(2ψ)(logQ). Replacing Ae=Q, this simplifies to the form (71).

7. Integrating the Lax pair system

It is clear that the first‐order equation in (67) for Q(s) is separable and can be integrated. The integral depends on whether K is zero or nonzero:

Q(s)=1εs+γ,K=0;log(2(K+KQ2+K)Q)=εKs+γ,K0.
(72)

Here ε=±1 and γ is the last constant of integration. Taking specific choices for the constants, for example, eγ=2K when K0 and a=K, the set of solutions (72) for Q(s) can be summarized below.

Dom(s)Q(s)Dom(s)Q(s)s>01ss<01s0<s<πaasin(as)πa<s<0asin(as)s>0asinh(as)s<0asinh(as)
(73)

It is presumed that other choices of the constants can be geometrically eliminated in favor of Eq. (73). The solutions (73) are then substituted back into linear system (67). The first equation in (67) implies that either

ψ0,modπ2;2ψssin(2ψ)=Q.
(74)

Substitute ψ0 into the second equation in (67). It implies that (logQ)s=Q and ψ=π/2 gives (logQ)s=Q. In both cases Q(s) is a solution which already appears in Eq. (73).

For the second case in Eq. (74), the equation can be put in the form

(log|tan(ψ)|)s=Q.

Integrating we have for some function y(t) to be determined,

tan(ψ)=eQ(s)dsy(t).
(75)

Therefore, tan(ψ) can be obtained by substituting for Q(s) for each of the three cases in Eq. (73). The upper sign holds for s>0 and the lower sign holds if s<0.

  1. Q(s)=±s1, Q(s)ds=log|s| and

    tan(ψ)=sy(t).
    (76)

  2. Q(s)=±asin(as), Q(s)ds=log|csc(as)cot(as)| and

    tan(ψ)=(tan(as2))y(t).
    (77)

  3. Q(s)=±asinh(as), Q(s)ds=arctanh(eas), and

    tan(ψ)=(tanh(as2))y(t).
    (78)

In case (ii), if s>0 and y(t)=±1 then ψ=±12(as+π), modπ, and if s<0 and y(t)=±1, then ψ=±12as, modπ.

It remains to integrate the second equation of the Lax pair (67) 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) Consider Q(s)=s1 and tan(ψ)=s1y(t). The second equation in (67) simplifies considerably to yt=1, therefore,

y(t)=(t+σ),tan(ψ)=(t+σ)s.
(79)

For Q(s)=s1 and tan(ψ)=sy(t), the second equation of (67) becomes yt=y2, therefore,

y(t)=1t+σ,tan(ψ)=st+σ.
(80)

8. A third‐order equation for Hand fundamental forms

Since ξ12=(loge(s))dt, using Eq. (60) ω12 can be written as

ω12=ξ12dψ=(loge(s))dtdψ.
(81)

Using Eqs. (14) and (64) for α1, it follows that

dlog(J)=Q(cos(2ψ)dssin(2ψ)dt)2*(ψtdt+ψsds)+2*(log(e(s)))dt.

when 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

dlog(J)=(Qcos(2ψ)+2ψt2(log(e(s))))ds+(Qsin(2ψ)2ψs)dt.
(82)

First‐order system (67) permits this to be written using e(s)=E(s) as

(log(J))+(log(E))=(log(Q)).
(83)

Hence, there exists a constant τ independent of s such that EJ=τQ or

E=τQJ=τQ2H.
(84)

This result (84) for E is substituted into the Gauss equation −((log(E))ss+(log(E))tt)=2E(H2J2) giving

(log(E))=2(log(Q))(log(Hs))=2Q2(HH).
(85)

Therefore, the Gauss equation transforms into a third‐order differential equation in the s variable,

(HH)+2τH=2Q2(1+τH2H).
(86)

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

I=E(s)(ds2+dt2),E(s)=τQ2(s)H(s).
(87)

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

II=Lds2+2Mdsdt+Ndt2.
(88)

where the coefficients L,M,N are given by

L=E(H+Jcos(2ψ))=EH+τQcos(2ψ),M=EJsin(2ψ)=τQsin(2ψ),N=E(HJcos(2ψ)).
(89)

Appendix

It is worth seeing how the second equation in (67) can be integrated for cases (ii) and (iii). Only the case s>0 will be done with Q(s) taken from Eq. (73).

(a) Differentiating tan(ψ) given in Eq. (77), we obtain that

ψt=tan(as2)tan2(as2)+y2yt(t).

The following identities are required to simplify the result,

tan(as)=2tan(as2)1tan2(as2),cos(2ψ)=tan2(as2)y2tan2(as2)+y2.

Substituting ψt into Eq. (67), we obtain

2tan(as2)tan2(as2)+y2yt=acot(as)asin(as)tan2(as2)y2tan2(as2)+y2.

Simplifying this, we get

4ayt=12(1tan2(as2))12(cot2(as2)1)y2sec2(as2)+csc2(as2)y2.

This simplifies to the elementary equation,

yt=a2(y21),y(t)=tanh(at2+η).

Here η is an integration constant. To summarize then,

tan(ψ)=tanh(at2+η)tan(as+π2).

(b) Consider now s>0 and take Q(s) from the last line of Eq. (73). Differentiating tan(ψ) from (78), we get

ψt=coth(as2)1+coth2(as2)y2yt(t).

In this case, the following identities are needed,

tanh(as)=2tanh(as2)1+tanh2(as2),cos(2ψ)=1coth2(as2)y21+coth2(as2)y2.

Therefore, Eq. (67) becomes

2coth(as2)1+coth2(as2)y2yt=acoth(as)asinh(as)tanh2(as2)y2tanh2(as2)+y2.

This reduces to

4ayt=(1+tanh2(as2)+sech2(as2))+(coth2(as2)+1csch2(as2))y2.

Simplifying and integrating, it has been found that

yt=a2(1+y2),y(t)=tan(at2+η).

To summarize then, it has been shown that,

tan(ψ)=cot(at2+η)coth(as2).

These results apply to the case s>0 and similar results can be found for the case s<0 as well.

MSCs: 53A05, 58A10, 53B05

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