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

Written By

Paul Bracken

Submitted: 21 November 2016 Reviewed: 22 November 2016 Published: 18 January 2017

DOI: 10.5772/67008

From the Edited Volume

Manifolds - Current Research Areas

Edited by Paul Bracken

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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.


  • 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


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


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


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


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


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


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


Define the variable J to be


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


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,


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,


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


Finally, note that


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



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


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


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


Proof: Exterior differentiation of ω1* produces


Similarly, differentiating ω2* gives


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


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


Substituting for ω12* from Theorem 3.1, this is


Lemma 3.1


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




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*=(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


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


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


Proof: Differentiating both sides of Lemma 3.3 gives


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


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


the associated connection forms are related by


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


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


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


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


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


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


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


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


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


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


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


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


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


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


The structure equation for the ϑi will be needed,


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


Using Eq. (31) in Eq. (30),


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


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


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


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


Differentiating structure equation (30) and using Lemma 4.1,




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


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


Theorem 5.1 The function ψ satisfies the equation


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


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


This can be put in the equivalent form


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


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


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


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


This linear system can be solved for C1 and C2 to get


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


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


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


Using Eq. (45), it follows that


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


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


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


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


Solving Eq. (26) for ω12, we have


The exterior derivative of this takes the form,


Putting this in the Gauss equation,


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


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


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

Lemma 5.2


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




Lemma 5.3




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


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


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,


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


Consequently, we can write


Therefore, it must be that


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


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,


The corresponding coframe and connection form are


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:


The condition dϑ1=0 is equivalent to


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,


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


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


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,


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


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),


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


Equating coefficients of differentials, this implies that


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


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


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


Therefore, H=AJe>0 and so H(s) is an increasing function of s. Now define the function Q(s) to be


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:


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


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


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


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


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

Differentiating both sides of Eq. (69) we get


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


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:


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.


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


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


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


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


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


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


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,


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



8. A third‐order equation for Hand fundamental forms

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


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


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


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


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


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


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


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


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


where the coefficients L,M,N are given by



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


The following identities are required to simplify the result,


Substituting ψt into Eq. (67), we obtain


Simplifying this, we get


This simplifies to the elementary equation,


Here η is an integration constant. To summarize then,


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


In this case, the following identities are needed,


Therefore, Eq. (67) becomes


This reduces to


Simplifying and integrating, it has been found that


To summarize then, it has been shown that,


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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Written By

Paul Bracken

Submitted: 21 November 2016 Reviewed: 22 November 2016 Published: 18 January 2017