Surfaces that admit isometries which preserve principal curvatures have been studied since the time of O. Bonnet  . It was shown by Bonnet that all surfaces with constant mean curvature not including planes and spheres, can be isometrically deformed while preserving mean curvature or equivalently both principal curvatures. Let denote the classes of smooth oriented, connected surfaces carrying a Riemannian metric which will be studied here. Let be a given smooth function, then depending on the surface and the function , it is possible to find isometric immersions of into such that each image has mean curvature function . It may be asked how many geometrically distinct, or noncongruent, immersions exists. This question has been studied at various levels both locally and globally. It has been concluded that the number of noncongruent immersions which is denoted here by , may be and . The case in which means there exists a one-parameter family of pairwise noncongruent isometric immersions with the same mean curvature function. When or , these surfaces are the so-called Bonnet surfaces. When or 1, there are many well-known surfaces, so all of these numbers can be realized.
Some of the global results that have been reported up to this point should be reviewed first. 1) If is constant and there is an isometric immersion of into of mean curvature which is not a plane or sphere, then there is an isometric deformation of through noncongruent surfaces with the same constant mean curvature, . 2) If is compact and is nonconstant, there exists at most two geometrically distinct immersions of in with mean curvature , so or 2. If is homeomorphic to , the 2-sphere, there is at most one isometric immersion, and is 0 or 1. 3) If is a helicoidal surface in , then or . 4) If the Gaussian curvature of is a nonzero constant, and the mean curvature is nonconstant, then or 2.
The surfaces in Euclidean space that admit a mean curvature preserving isometry which is not an isometry of the whole space form a special class of surface which has been studied by many people such as noted already Bonnet as well as Cartan and Chern    . These surfaces may be broken up into three classes or types which can be described as follows: 1) There are surfaces of constant mean curvature other than the plane or sphere 2) There are certain surfaces of nonconstant mean curvature which admit a one-parameter family of geometrically distinct nontrivial isometries, and finally 3) There are surfaces of nonconstant mean curvature that admit a single nontrivial isometry which is unique up to an isometry of the entire space.
A surface that belongs to one of the above types is called a Bonnet surface, that is, an type mentioned above    . By a nontrivial isometry of the surface is meant an isometry of the entire space. A helicoidal surface in Euclidean three-space is the locus of an appropriately chosen curve under a helicoidal motion, with so-called pitch in the interval   . Such a motion can be described by a one-parameter group of isometries in . The actual orbits of the motion through the initial curve foliate the surface.
The intention here is to prove that the helicoidal surfaces are necessarily Bonnet surfaces, and moreover represent all three types of surface outlined above. Although not all the theorems presented here are new, the objective is to present new proofs based on the systematic use of differential forms and the moving frame concept  . This type of result is useful to have since it provides an answer, in the negative, to conjectures such as the following: Let be a Riemannian surface and a smooth function. If a nontrivial family of isometric immersions with mean curvature function does not exist, then there must be at most two noncongruent ones. Then it may be conjectured: In the absence of a nontrivial family, the immersion must be unique. On the other hand, it seems that not all Bonnet surfaces of the third type are helicoidal surfaces. A helicoidal surface is determined by one real-valued function of one variable, whereas a Bonnet surface of the third kind depends on four functions of one variable and therefore has a greater degree of generality  .
2. Structure Equations
Over there exists a system of orthonormal frames which is well defined such that , is the unit normal at and , located along principal directions. The fundamental equations for a surface have the form  ,
These equations can be differentiated exteriorly in turn and 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 at which are denoted by and by writing
It suffices to suppose that in the following and the mean curvature of will be denoted by and the Gaussian curvature is denoted by . They are defined in terms of the functions and to be
The forms which appear in (2.1) satisfy the fundamental set of structure equations
The second pair of equations in (2.4) is referred to as the Codazzi equation and the last equation is called the Gauss equation.
Exterior differentiation of the Codazzi equations in (2.4) and using (2.2) yields
Now Cartan’s lemma can be applied to (2.5). There exist two functions and such that
Subtracting the pair of equations in (2.6) gives an expression for ,
It is natural from (2.7) to define a new variable in terms of and Math_76# as
Equation (2.7) can then be put in the form,
The differential forms constitute a linearly independent system. Two related coframes and can be defined in terms of the and the functions and as follows
These relations imply that is tangent to the level curves which are singled out by setting equal to a constant and is its symmetry with respect to the original directions.
The relation is squared and subtracting the definition of the Gaussian curvature, yields the result . The Hodge operator, denoted here by , will play an important role in the following. It produces the following result on the basis forms in (2.2),
From these properties, the dual relations can be determined as
Moreover, adding the expressions for and given by (2.6), we obtain
Finally, there is the relation,
Therefore, the Codazzi Equations (2.12) and (2.13) can be summarized in terms of the two functions and as follows,
3. Bonnet Surfaces
Suppose that is a surface which is isometric to such that the principal curvatures are preserved under the transformation. Denote all quantities which pertain to by the same symbols, but with an asterisk,
The same convention will be applied to the variables and forms which pertain to and . When and are isometric, the forms on are related to the forms on by means of the transformation
The following theorem from  will be required.
Theorem 3.1: Under the transformation of coframe given in (3.2), the associated connection forms are related by
There is a very important result which can be developed at this point. In the case that and , the Codazzi equations imply that
Now apply the operator to both sides of this equation to give,
Substituting for from Theorem 3.1, this assumes the form,
Define and to be the derivatives of the function in directions such that can be expanded as
Since and using given by (2.10), Equation (2.13) produces
Comparing coefficients of and on both sides, we can identify
This result implies that
Since , it follows from (3.4) that
Solving this for and substituting for and , it follows that
Therefore, using the second equation of (2.14) for implies that
The differential in (3.6) will play a role in the study of helicoidal surfaces.
4. Helicoidal Surfaces
Every helicoidal surface can be parametrized in terms of two parameters , where can be thought of as time along orbits from a fixed , and is an arc-length of curves orthogonal to orbits. Then the curves are carried along the orbits by the helicoidal motion for constant. They remain orthogonal to the orbits and foliate the surface. An orthonormal frame , is determined along these coordinate curves. The corresponding coframe may be written as
where depends only on . Since , the equation implies that is proportional to , say and implies that
Hence, the -curves are geodesics, and the -curves or orbits, have geodesic curvature equal to
Thus along each orbit, the quantities , , and are constant and depend only on . In this case, the derivative . Also for the same reason, the differential form of implies that , hence
Hence, the equation for in (3.6) takes the form
Writing as a differential form in terms of and and then equating coefficients of and on both sides of (4.5) yields the following pair of equations,
The results in (4.6) are used in the proof of Theorem 5.2 which follows.
5. Main Theorems and Proofs
Now by what has been established so far, both functions and depend on the variable , so this mapping is an isometry which preserves , since is the average of and . In general, an isometry is trivial if and only if it preserves the mean curvature and the principal directions. In this case, the above mapping is trivial if and only if is a multiple of . Then we obtain that the orbits are plane curves. But this is impossible for a helicoidal surface. This proves the following result.
Theorem 5.1. For a helicoidal surface, the mapping is a nontrivial isometry which preserves the mean curvature .
To prove the second theorem, the following result due to Chern is required   .
Proposition 5.1. (Chern) A surface admits a nontrivial isometric deformation that keeps the principal curvatures fixed if and only if
Theorem 5.2. A helicoidal surface is a Bonnet surface of the second type if and only if the following relation is satisfied,
Proof: Set and consider the principal coframe
Define and as the coefficients in the differential by putting and let and be given by (3.5). Next we substitute and into the equations which appear in Chern’s result given in Proposition 5.1. Since in the basis of forms, it follows that
Since the first equation is , this implies that
Similarly, using (3.5), we have
Equating these two results as in the second of Chern’s two equations, we obtain
Multiplying (5.3) by and (5.4) by , it is found that the folowing hold:
Adding these two equations, the desired result is obtained,
Replacing by in (5.5), equation (5.1) follows.
Multiply (5.5) by to obtain,
Substituting the derivative for into (5.6), it becomes,
By means of the product rule, this can be put in the form,
This is trivial to integrate, so if is the integration constant, we obtain that
Since this relation may be viewed as an ordinary differential equation for the real-valued function which determines the helicoidal surface under helicoidal motion, the existence of such a surface is guaranteed by the local existence uniqueness theorem for solutions of such an ordinary differential equation.
From the first equation of (4.6) and the fact that the space curvature of orbits
is either or this with interchanged,
the last result follows.
Theorem 5.3. A helicoidal surface has constant mean curvature if and only if its principal directions make an angle constant with the orbits.
Combining all of these results, the main result of this work can be stated in the form of the following Theorem.
Theorem 5.4. The helicoidal surfaces are necessarily Bonnet surfaces and they represent all three types of surface.
A conclusion that follows from these results then is an interesting new geometric characterization of such surfaces. Thus, a helicoidal surface has constant mean curvature if and only if its principal directions make an angle which is constant with the orbits.
Finally, it will be proved that for any surface of revolution in which has nonconstant mean curvature function it holds that either or .
Let be a plane curve in the plane and form the surface of revolution
The principal curvatures are calculated to be
If at , of course the entire parallel through consists of umbilic points, so . Here denotes the derivative of with respect to .
Theorem 5.5: Surfaces of revolution with nonconstant mean curvature that admit a one-parameter family of geometrically distinct nontrivial isometries preserving principal curvatures are exactly those for which the function satisfies a specific fourth order differential equation in .
Proof: For the surface of revolution of the form (5.9), the principal coframe is given by
Since principal curvatures and in (5.10) depend only on and not on , the first equation of (2.14) implies that
Equating the coefficients of the differentials and on both sides gives and ,
Then the forms and can be calculated from (2.10),
Substituting (5.14) into the differential expressions of Proposition 5.1, it is clear that must always hold since the coefficient of depends only on and . To develope the second equation of the pair, we calculate
Equating these two expressions, the following fourth-order differential equa- tion for is obtained,
In (5.16), the principal curvatures and are given in (5.10), and since contains second derivatives of with respect to , equation (5.16) will be a fourth order equation in . This is the equation mentioned in the Theorem.
 Chern, S.S. (1955) An Elementary Proof of the Existence of Isothermal Parameters on a Surface. Proceedings of the American Mathematical Society, 6, 771-782.
 Chern, S.S. (1985) Deformation of Surfaces Preserving Principal Curvatures. In: Rauch, H.E., Ed., Differential Geometry and Complex Analysis, Springer, Berlin, 155-163.
 Bracken, P. (2017) An Intrinsic Characterization of Bonnet Surfaces Based on a Closed Differential Ideal. Manifolds-Current Research Areas, ed. P. Bracken, Intech Open Science Press, 21-43.