1. Introduction
Surfaces that admit isometries which preserve principal curvatures have been studied since the time of O. Bonnet [1] . 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 [2] [3] [4] . 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 [5] [6] [7] . 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
[8] [9] . 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 [7] . 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 [10] .
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 [11] ,
(2.1)
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
(2.2)
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
(2.3)
The forms which appear in (2.1) satisfy the fundamental set of structure equations
(2.4)
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
(2.5)
Now Cartan’s lemma can be applied to (2.5). There exist two functions
and
such that
(2.6)
Subtracting the pair of equations in (2.6) gives an expression for
,
(2.7)
It is natural from (2.7) to define a new variable
in terms of
and
Math_76# as
(2.8)
Equation (2.7) can then be put in the form,
(2.9)
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
(2.10)
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
(2.11)
Moreover, adding the expressions for
and
given by (2.6), we obtain
(2.12)
Finally, there is the relation,
(2.13)
Therefore, the Codazzi Equations (2.12) and (2.13) can be summarized in terms of the two functions
and
as follows,
(2.14)
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,
(3.1)
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
(3.2)
The following theorem from [7] will be required.
Theorem 3.1: Under the transformation of coframe given in (3.2), the associated connection forms are related by
(3.3)
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,
(3.4)
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
(3.5)
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
(3.6)
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
(4.1)
where
depends only on
. Since
, the equation
implies that
is proportional to
, say
and
implies that
so,
(4.2)
Hence, the
-curves are geodesics, and the
-curves or orbits, have geodesic curvature equal to
(4.3)
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
(4.4)
Hence, the equation for
in (3.6) takes the form
that is,
(4.5)
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,
(4.6)
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 [4] [7] .
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,
(5.1)
with
nonconstant.
Proof: Set
and consider the principal coframe
(5.2)
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
(5.3)
Similarly, using (3.5), we have
Equating these two results as in the second of Chern’s two equations, we obtain
(5.4)
Multiplying (5.3) by
and (5.4) by
, it is found that the folowing hold:
Adding these two equations, the desired result is obtained,
(5.5)
Replacing
by
in (5.5), equation (5.1) follows.
Multiply (5.5) by
to obtain,
(5.6)
Substituting the derivative for
into (5.6), it becomes,
By means of the product rule, this can be put in the form,
(5.7)
This is trivial to integrate, so if
is the integration constant, we obtain that
(5.8)
with
nonconstant.
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
(5.9)
The principal curvatures are calculated to be
(5.10)
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
(5.11)
Since principal curvatures
and
in (5.10) depend only on
and not on
, the first equation of (2.14) implies that
(5.12)
Equating the coefficients of the differentials
and
on both sides gives
and
,
(5.13)
Then the forms
and
can be calculated from (2.10),
(5.14)
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
(5.15)
Equating these two expressions, the following fourth-order differential equa- tion for
is obtained,
(5.16)
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.