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.
- differential form
- differential equation
- Bonnet surface
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 . 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  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 in with nonconstant mean curvature function . Moreover, suppose admits infinitely many nontrivial and geometrically distinct isometries preserving . Suppose is the set of umbilic points of and the set of critical points of . Many global facts are known with regard to and , and a few will now be mentioned. The set consists of isolated points, even if there exists only one nontrivial isometry preserving the mean curvature, moreover, [7, 8]. Interestingly, there is no point in at which all order derivatives of are zero, and 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 continuously defined [9–11]. As first noted by Chern , this function is harmonic. The analysis will begin by formulating the structure equations for two‐dimensional manifolds.
2. Structure equations
Over , there exists a well‐defined field of orthonormal frames, which is written as , such that , is the unit normal at , and are along principal directions . The fundamental equations for have the form
Differentiating each of these equations in turn, results in a large system of equations for the exterior derivatives of the and , as well as a final equation which relates some of the forms . This choice of frame and Cartan's lemma allows for the introduction of the two principal curvatures which are denoted by and at by writing
Suppose that in the following. The mean curvature of is denoted by and the Gaussian curvature by . They are related to and as follows
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 and such that
Subtracting the pair of equations in (6) gives an expression for
Define the variable to be
It will appear frequently in what follows. Equation (7) then takes the form
The constitute a linearly independent set. Two related coframes called and can be defined in terms of the and the functions and as follows,
These relations imply that is tangent to the level curves specified by equals constant and is its symmetry with respect to the principal directions.
Squaring both sides of the relation and subtracting the relation yields . The Hodge operator, denoted by , will play an important role throughout. It produces the following result on the basis forms ,
Moreover, adding the expressions for and given in Eq. (6), there results
Finally, note that
3. A theorem of Bonnet
Suppose that is a surface which is isometric to such that the principal curvatures are preserved [10–12]. Denote all quantities which pertain to with the same symbols but with asterisks, as for example
The same notation will be applied to the variables and forms which pertain to and . When and are isometric, the forms are related to the by the following transformation
Similarly, differentiating gives
There is a very important result that can be developed at this point. In the case that and , the Codazzi equations imply that
Apply the operator to both sides of this equation, we obtain
Substituting for from Theorem 3.1, this is
where and .
Lemma 3.1 also follows from the fact that and Eq. (8).
Substituting from Lemma 3.2 into Eq. (13), can be written as
Introduce the new variable so and , , hence the following lemma.
This is the total differential equation which must be satisfied by the angle 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.
Equating the coefficients of to zero gives the result (20).
This theorem seems to originate with Chern  and is very useful because it gives the exterior derivatives of the . When the mean curvature is constant, , hence it follows from Eq. (14) that . This implies that , and so and must vanish. Hence, which implies that, since the is linearly independent, equals a constant. Thus, we arrive at a theorem originally due to Bonnet.
4. Connection form associated to a coframe and transformation properties
Given the linearly independent one forms , the first two of the structure equations uniquely determine the form . The is called the orthonormal coframe of the metric
and is the connection form associated with it.
the associated connection forms are related by
Using Eq. (21) to replace the in these, we obtain
The satisfy a similar system of structure equations, so replacing here yields
Since the form satisfies the equations and , substituting these relations into the above equations and using , we obtain that in the form
Cartan's lemma can be used to conclude from these that there exist functions and such that
Finally, apply to both sides and use to obtain
The forms are linearly independent, so for these two equations to be compatible, it suffices to put , and the result follows.
For the necessity in the Chern criterion, Theorem 3.2, no mention of the set of critical points of is needed. In fact, when is constant, this criterion is met and the sufficiency also holds with constant. However, when is not identically constant, we need to take the set of critical points into account for the sufficiency. In this case, is also an open, dense, and connected subset of . On this subset and the function can be defined in terms of the functions and as
To define more general transformations of the , define the angle as
This angle, which is defined modulo , is continuous only locally and could be discontinuous in a nonsimply connected region of . With and related to and by Eq. (24), the forms and can be written in terms of and as
The forms , , define the same structure on and we let , , be the connection forms associated to the coframes ; ; . The next theorem is crucial for what follows.
The transformation to the is exactly similar except that , hence
This implies . When replaced in the first equation of (26), the second equation appears. Note that from Theorem 3.2, , so the second equation can be given as .
Differentiating the second equation in Eq. (14) and using , it follows that
Exterior differentiation of this equation using immediately gives
This states that is a harmonic function. Equation (28) also implies that .
5. Construction of the closed differential ideal associated with
Exterior differentiation of the first equation in (14) and using the second equation produces
The structure equation for the will be needed,
Replacing by means of Eq. (29) implies the following important result
Equation (33) and Cartan's lemma imply that there exists a function such that
This is the first in a series of results which relates many of the variables in question such as , , and directly to the one‐form . To show this requires considerable work. The way to proceed is to use the forms in Theorem 3.2 because their exterior derivatives are known. For an arbitrary function on , define
Differentiating Eq. (35) and extracting the coefficient of , we obtain
In terms of the , , Lemma 4.1 yields
Finally, since , substituting for , we obtain that
Differentiating structure equation (30) and using Lemma 4.1,
This equation implies that either or is a multiple by a function of the form . Hence, for some function ,
Substituting the first line of Eq. (39) back into the structure equation, we have
This can be put in the equivalent form
Taking the exterior product with and using , we get
Imposing the constraint (41), the coefficient of can be equated to zero. This produces the result (42).
As a consequence of Theorem 5.1, a new function can be introduced such that
Differentiation of each of these with respect to the basis, we get for that
This linear system can be solved for and to get
By differentiating each of the equations in (46), it is easy to verify that satisfies Eq. (36), namely, . Hence, there exist harmonic functions which satisfy Eq. (42). The solution depends on two arbitrary constants, the values of and at an initial point.
Using Eq. (45), it follows that
This implies that .
Differentiating each of these in turn, we obtain for ,
Taking in Eq. (36) produces a first equation for the ,
If another equation in terms of and 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 , 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
Given these results for and , it is easy to produce the following two Lemmas.
In the interests of completeness, it is important to verify the following theorem.
Note that the coefficient of in this appears in the compatibility condition. To express it in another way, begin by finding the exterior derivative of ,
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 , 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
During the prolongation of the exterior differential system, the additional variables , , , and have been introduced. The significance of the appearance of the function , 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.
Along the , curves, consider the normalized frame,
The corresponding coframe and connection form are
Then can be expressed as a multiple of and in terms of , and the differential system can be summarized here:
The condition is equivalent to
This implies that since is proportional to . Also, is equivalent to .
Moreover, is equivalent to the fact that the curves can be regarded as coordinate curves parameterized by isothermal parameters. Therefore, along the curves, orthogonal isothermal coordinates denoted can be introduced. The first fundamental form of then takes the form,
Now suppose we set , then
This means such a surface is isometric to a surface of revolution. Since , , Eq. (57) implies that . This can be stated otherwise as the principal coordinates are isothermal and so is an isothermic surface.
Since , and are functions of only the variable , this implies that and , or and , are constant along the curves where is constant. This leads to the following proposition.
This is equivalent to the statement is a Weingarten surface.
Consequently, the geodesic curvature of each curve, constant, is
which is constant.
To express the in terms of and , start by writing in terms of the and then substituting Eq. (60),
Subscripts denote differentiation and is used interchangeably. Beginning with and using Eq. (62), we have
Equating coefficients of differentials, this implies that
Solving this as a linear system we obtain , ,
Noting that and , using Eq. (57) the forms can be expressed in terms of
Substituting from Eq. (60) into ,
Therefore, and so is an increasing function of . Now define the function to be
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 is harmonic as well. Differentiating with respect to and with respect to , it is clear that satisfies Laplace's equation in the variables . This is another proof that is harmonic.
There exists a first integral for this equation of the following form
Equating derivatives , the required (68) follows.
Differentiating both sides of Eq. (69) we get
It is important to note that the function which appears when the differential ideal closes can be related to the function .
7. Integrating the Lax pair system
It is clear that the first‐order equation in (67) for is separable and can be integrated. The integral depends on whether is zero or nonzero:
Here and is the last constant of integration. Taking specific choices for the constants, for example, when and , the set of solutions (72) for 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
For the second case in Eq. (74), the equation can be put in the form
Integrating we have for some function to be determined,
Therefore, can be obtained by substituting for for each of the three cases in Eq. (73). The upper sign holds for and the lower sign holds if .
, , and
In case , if and then , , and if and , then , .
It remains to integrate the second equation of the Lax pair (67) using solutions for both and . The first case is not hard and will be shown explicitly here. The others can be done, and more complicated cases are considered in the Appendix.
Consider and . The second equation in (67) simplifies considerably to , therefore,
For and , the second equation of (67) becomes , therefore,
8. A third‐order equation for Hand fundamental forms
Since , using Eq. (60) can be written as
when are put in the coordinates, using , it can be stated that and . Consequently, simplifies to
First‐order system (67) permits this to be written using as
Hence, there exists a constant independent of such that or
This result (84) for is substituted into the Gauss equation −((
Therefore, the Gauss equation transforms into a third‐order differential equation in the variable,
Thus, a characterization of Bonnet surfaces is reached by means of the solutions to these equations. This equation determines the function and after that the functions and . Therefore, Bonnet surfaces have as first fundamental form the expression
Since is the angle from the principal axis to the curve with equals constant, the second fundamental form is given by
where the coefficients are given by
Differentiating given in Eq. (77), we obtain that
The following identities are required to simplify the result,
Substituting into Eq. (67), we obtain
Simplifying this, we get
This simplifies to the elementary equation,
Here is an integration constant. To summarize then,
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 and similar results can be found for the case as well.
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