2. Structure equations

Over M, there exists a well-defined field of orthonormal frames, which is written as x, e1;e2;e<sup>3</sup> such that x∈M, e<sup>3</sup> is the unit normal at x, and e1;e<sup>2</sup> are along principal directions [12]. The fundamental equations for M have the form

$$d\mathbf{x} = \omega\_1 \mathbf{e}\_1 + \omega\_2 \mathbf{e}\_2, \quad d\mathbf{e}\_1 = \omega\_{12} \mathbf{e}\_2 + \omega\_{13} \mathbf{e}\_3, \quad d\mathbf{e}\_2 = -\omega\_{12} \mathbf{e}\_1 + \omega\_{23} \mathbf{e}\_3, \quad d\mathbf{e}\_3 = -\omega\_{13} \mathbf{e}\_1 - \omega\_{23} \mathbf{e}\_2. \tag{1}$$

Differentiating each of these equations in turn, results in a large system of equations for the exterior derivatives of the ω<sup>i</sup> 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

$$
\omega\_{12} = h\omega\_1 + k\omega\_2, \qquad \omega\_{13} = a\omega\_1, \qquad \omega\_{23} = c\omega\_2. \tag{2}
$$

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

$$H = \frac{1}{2}(a+c), \qquad K = a \cdot c. \tag{3}$$

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

An Intrinsic Characterization of Bonnet Surfaces Based on a Closed Differential Ideal http://dx.doi.org/10.5772/67008 23

(4)

$$d\omega\_1 = \omega\_{12}\wedge\omega\_2,\tag{4} \qquad\qquad\qquad d\omega\_2 = \omega\_1\wedge\omega\_{12}$$

$$d\omega\_{13} = \omega\_{12}\wedge\omega\_{23}\tag{5} \qquad\qquad\qquad\qquad\qquad\qquad d\omega\_{23} = \omega\_{13}\wedge\omega\_{12},\tag{6}$$

$$d\omega\_{12} = ac\,\,\omega\_2 \wedge \omega\_1 = -K\,\,\omega\_1 \wedge \omega\_2.$$

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

$$(da - (a \text{-c}) h\omega\_2) \wedge \omega\_1 = 0, \qquad (dc - (a \text{-c}) k\omega\_1) \wedge \omega\_2 = 0. \tag{5}$$

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

$$\frac{1}{a \multimap c} da \text{--} h\omega\_2 = (u \multimap k)\omega\_1, \qquad \frac{1}{a \multimap c} d\text{-} k\omega\_1 = (v \multimap l)\omega\_2 \tag{6}$$

Subtracting the pair of equations in (6) gives an expression for dlogða−cÞ

$$d\log(a \cdot c) = (\mu \text{--} 2k) \,\, \omega\_1 \text{--} (\upsilon \text{--} 2h) \,\, \omega\_2. \tag{7}$$

Define the variable J to be

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

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 intercon-

To establish some information about what is known, consider an oriented, connected, smooth open surface M in E<sup>3</sup> 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, U⊂V [7, 8]. Interestingly, there is no point in V−U 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 [9–11]. As first noted by Chern [6], this function is harmonic. The analysis will begin

Over M, there exists a well-defined field of orthonormal frames, which is written as x, e1;e2;e<sup>3</sup> such that x∈M, e<sup>3</sup> is the unit normal at x, and e1;e<sup>2</sup> are along principal directions [12]. The

dx ¼ ω1e<sup>1</sup> þ ω<sup>2</sup> e2; de<sup>1</sup> ¼ ω12e<sup>2</sup> þ ω13e3; de<sup>2</sup> ¼ −ω12e<sup>1</sup> þ ω23e3; de<sup>3</sup> ¼ −ω13e1−ω23e2: (1)

ω<sup>12</sup> ¼ hω<sup>1</sup> þ kω2; ω<sup>13</sup> ¼ aω1; ω<sup>23</sup> ¼ cω2: (2)

ða þ cÞ; K ¼ a � c: (3)

Differentiating each of these equations in turn, results in a large system of equations for the exterior derivatives of the ω<sup>i</sup> 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

Suppose that a > c in the following. The mean curvature of M is denoted by H and the

The forms which appear in Eq. (1) satisfy the fundamental structure equations which are

by formulating the structure equations for two-dimensional manifolds.

satisfy nontrivial ordinary differential equations.

22 Manifolds - Current Research Areas

nectedness of the ideas and results involved.

2. Structure equations

summarized here [14],

fundamental equations for M have the form

curvatures which are denoted by a and c at x by writing

Gaussian curvature by K. They are related to a and c as follows

<sup>H</sup> <sup>¼</sup> <sup>1</sup> 2

$$J = \frac{1}{2}(a \text{-} c) > 0. \tag{8}$$

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

$$d\log \mathcal{J} = (\mu - 2k)\omega\_1 - (\upsilon - 2h)\omega\_2. \tag{9}$$

The ω<sup>i</sup> constitute a linearly independent set. Two related coframes called ϑ<sup>i</sup> and α<sup>i</sup> can be defined in terms of the ω<sup>i</sup> and the functions u and v as follows,

$$\begin{aligned} \mathfrak{G}\_1 &= \mathfrak{u}\omega\_1 + \mathfrak{v}\omega\_2, & \mathfrak{G}\_2 &= -\mathfrak{v}\omega\_1 + \mathfrak{u}\omega\_2, \\\\ \mathfrak{a}\_1 &= \mathfrak{u}\omega\_1 \text{--} \mathfrak{v}\omega\_2, & \mathfrak{a}\_2 &= \mathfrak{v}\omega\_1 + \mathfrak{u}\omega\_2. \end{aligned} \tag{10}$$

These relations imply that ϑ<sup>1</sup> ¼ 0 is tangent to the level curves specified by H equals constant and α<sup>1</sup> ¼ 0 is its symmetry with respect to the principal directions.

Squaring both sides of the relation 2H ¼ a þ c and subtracting the relation 4K ¼ 4ac yields <sup>4</sup>ðH<sup>2</sup> −KÞ¼ða−cÞ 2 . The Hodge operator, denoted by �, will play an important role throughout. It produces the following result on the basis forms ωi,

$$\*\omega\_1 = \omega\_2, \qquad \*\omega\_2 = \neg\omega\_1, \qquad \*^2 = -1. \tag{11}$$

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

$$\frac{1}{a-c}(da+d\varepsilon) = (u-k)\omega\_1 + h\omega\_2 + (v-k)\omega\_2 + k\omega\_1 = u\omega\_1 + v\omega\_2 = \mathfrak{d}\_1.\tag{12}$$

Finally, note that

$$\alpha\_1 + \mathcal{D} \ast \omega\_{12} = \mathfrak{u}\omega\_1 \text{--} \mathfrak{v}\omega\_2 + \mathcal{D} \ast (h\omega\_1 + k\omega\_2) = (\mathfrak{u} - 2k)\omega\_1 \text{--} (\mathfrak{v} - 2h)\omega\_2 = d\text{log}I.\tag{13}$$

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

$$dH = I\\$\_1, \qquad d\text{log}I = \alpha\_1 + 2 \,\,\ast \,\omega\_{12.} \tag{14}$$
