3. A theorem of Bonnet

Suppose that M� is a surface which is isometric to M such that the principal curvatures are preserved [10–12]. Denote all quantities which pertain to M� with the same symbols but with asterisks, as for example

$$a^\* = a, \qquad c^\* = c.$$

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 ω<sup>i</sup> are related to the ω� <sup>i</sup> by the following transformation

$$
\omega\_1^\* = \cos \tau \,\,\omega\_1 - \sin \tau \,\,\omega\_2, \qquad \omega\_2^\* = \sin \tau \,\,\omega\_1 + \cos \tau \,\,\omega\_2. \tag{15}
$$

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

$$
\omega\_{12}^\* = \omega\_{12} \text{--} d\tau. \tag{16}
$$

Proof: Exterior differentiation of ω� <sup>1</sup> produces

> dω� <sup>1</sup> ¼ − sin τ dτ∧ω<sup>1</sup> þ cos τ dω1− cos τ dτ∧ω2− sin τ dω<sup>2</sup> ¼ dτ∧ð− sin τω1− cos τω2Þ þ cos τ ω12∧ω2− sin τω1∧ω<sup>12</sup> ¼ ð−dτ þ ω12Þ∧ω� 2:

Similarly, differentiating ω� <sup>2</sup> gives

$$\begin{aligned} d\omega\_2^\* &= \cos\tau\wedge\omega\_1 + \sin\tau\,d\omega\_1 - \sin\tau\,d\tau\wedge\omega\_2 + \cos\tau\,d\omega\_2\\ &= d\tau\wedge(\cos\tau\omega\_1 - \sin\tau\omega\_2) + \sin\tau\omega\_{12}\wedge\omega\_2 + \cos\tau\omega\_1\wedge\omega\_{12} = \omega\_1^\*\wedge(-d\tau + \omega\_{12}). \end{aligned}$$

There is a very important result that can be developed at this point. In the case that a ¼ a� and c ¼ c�, the Codazzi equations imply that

$$
\alpha\_1 + 2 \ast \omega\_{12} = d \log(a \cdot c) = d \log(a^\ast \neg c^\ast) = \alpha\_1^\ast + 2 \ast \omega\_{12}^\ast.
$$

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

$$
\alpha\_2 \text{--} 2a\nu\_{12} = \alpha\_2^\* \text{--} 2a\nu\_{12}^\*.
$$

Substituting for ω� <sup>12</sup> from Theorem 3.1, this is

$$
\Delta d \tau = \alpha\_2 \cdots \alpha\_2^\*.\tag{17}
$$

Lemma 3.1

�ω<sup>1</sup> <sup>¼</sup> <sup>ω</sup>2; �ω<sup>2</sup> <sup>¼</sup> <sup>−</sup>ω1; �<sup>2</sup> <sup>¼</sup> <sup>−</sup>1: (11)

dH ¼ Jϑ1; dlogJ ¼ α<sup>1</sup> þ 2 � ω<sup>12</sup>: (14)

ðda þ dcÞ¼ðu−kÞω<sup>1</sup> þ hω<sup>2</sup> þ ðv−kÞω<sup>2</sup> þ kω<sup>1</sup> ¼ uω<sup>1</sup> þ vω<sup>2</sup> ¼ ϑ1: (12)

α<sup>1</sup> þ 2 � ω<sup>12</sup> ¼ uω1−vω<sup>2</sup> þ 2 � ðhω<sup>1</sup> þ kω2Þ¼ðu−2kÞω1−ðv−2hÞω<sup>2</sup> ¼ dlogJ: (13)

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

Suppose that M� is a surface which is isometric to M such that the principal curvatures are preserved [10–12]. Denote all quantities which pertain to M� with the same symbols but with

� ¼ c:

a� ¼ a; c

When M and M� are isometric, the forms ω<sup>i</sup> are related to the ω�

<sup>1</sup> ¼ cos τ ω1− sin τ ω2; ω�

ω�

Proof: Exterior differentiation of ω�

Similarly, differentiating ω�

dω�

dω�

<sup>2</sup> gives

The same notation will be applied to the variables and forms which pertain to M and M�

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

¼ dτ∧ð− sin τω1− cos τω2Þ þ cos τ ω12∧ω2− sin τω1∧ω<sup>12</sup> ¼ ð−dτ þ ω12Þ∧ω�

<sup>1</sup> ¼ − sin τ dτ∧ω<sup>1</sup> þ cos τ dω1− cos τ dτ∧ω2− sin τ dω<sup>2</sup>

ω�

<sup>1</sup> produces

<sup>2</sup> ¼ cos τ∧ω<sup>1</sup> þ sin τ dω1− sin τ dτ∧ω<sup>2</sup> þ cos τ dω<sup>2</sup> ¼ dτ∧ð cos τω1− sin τω2Þ þ sin τω12∧ω<sup>2</sup> þ cos τω1∧ω<sup>12</sup> ¼ ω�

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

1 a−c

24 Manifolds - Current Research Areas

3. A theorem of Bonnet

asterisks, as for example

forms are related by

Finally, note that

and J as

tion

$$
\vartheta\_1 = \vartheta\_1^\*.
$$

Proof: This can be shown in two ways. First from Eq. (15), express the ω<sup>i</sup> in terms of the ω� i

$$
\omega\_1 = \cos \tau \,\omega\_1^\* + \sin \tau \,\omega\_2^\*, \qquad \omega\_2 = -\sin \tau \,\omega\_1^\* + \cos \tau \,\omega\_2^\*. \tag{18}
$$

Therefore,

.

<sup>i</sup> by the following transforma-

2:

<sup>2</sup> ¼ sin τ ω<sup>1</sup> þ cos τ ω2: (15)

<sup>1</sup>∧ð−dτ þ ω12Þ:

<sup>12</sup> ¼ ω12−dτ: (16)

$$
\vartheta\_1 = \mathfrak{u}\omega\_1 + \mathfrak{v}\omega\_2 = \mathfrak{u}(\cos \tau \,\omega\_1^\* + \sin \tau \omega\_2^\*) + \mathfrak{v}(-\sin \tau \omega\_1^\* + \cos \tau \omega\_2^\*) = \mathfrak{u}^\*\omega\_1^\* + \mathfrak{v}^\*\omega\_2^\* = \mathfrak{d}\_1^\*,
$$

where <sup>u</sup>� <sup>¼</sup> <sup>u</sup> cos <sup>τ</sup>−<sup>v</sup> sin <sup>τ</sup> and <sup>v</sup>� <sup>¼</sup> <sup>u</sup> sin <sup>τ</sup> <sup>þ</sup> <sup>v</sup> cos <sup>τ</sup>. □

Lemma 3.1 also follows from the fact that dH ¼ dH� and Eq. (8).

#### Lemma 3.2

$$
\alpha\_2^\* = \sin\left(2\pi\right)\alpha\_1 + \cos\left(2\pi\right)\alpha\_2.
$$

Proof:

$$\begin{array}{l} a\_2^\* = (\iota \sin \tau + v \cos \tau)(\cos \tau \omega\_1 - \sin \tau \omega\_2) + (\iota \cos \tau - v \sin \tau)(\sin \tau \omega\_1 + \cos \tau \omega\_2) \\ = (\iota \sin \left(2\tau\right) + v \cos \left(2\tau\right)) \omega\_1 + (-v \sin \left(2\tau\right) + \iota \cos \left(2\tau\right)) \omega\_2 \\ = \sin \left(2\tau\right) a\_1 + \cos \left(2\tau\right) a\_2. \end{array}$$

Substituting α� <sup>2</sup> from Lemma 3.2 into Eq. (13), dτ can be written as

$$d\tau = \frac{1}{2} (a\_2 - \sin\left(2\pi\right)a\_1 - \cos\left(2\pi\right)a\_2) = \frac{1}{2} ((1 - \cos\left(2\pi\right))a\_2 - \sin\left(2\pi\right)a\_1). \tag{19}$$

Introduce the new variable <sup>t</sup> <sup>¼</sup> cotðτ<sup>Þ</sup> so dt <sup>¼</sup> <sup>−</sup>csc2ðτ<sup>Þ</sup> <sup>d</sup><sup>τ</sup> and sin <sup>τ</sup> <sup>¼</sup> <sup>1</sup>ffiffiffiffiffiffiffi 1þt 2 <sup>p</sup> , cos <sup>τ</sup> <sup>¼</sup> <sup>1</sup>ffiffiffiffiffiffiffi 1þt 2 p , hence the following lemma.

Lemma 3.3 dt <sup>¼</sup> <sup>t</sup>α1−α2:

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

$$d\alpha\_1 = 0, \qquad d\alpha\_2 = \alpha\_1 \wedge \alpha\_2. \tag{20}$$

Proof: Differentiating both sides of Lemma 3.3 gives

$$t dt \wedge \alpha\_1 + t d\alpha\_1 - d\alpha\_2 = (t \alpha\_1 - \alpha\_2) \wedge \alpha\_1 + t d\alpha\_1 - d\alpha\_2 = 0.1$$

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 ¼ 0, hence it follows from Eq. (14) that ϑ<sup>1</sup> ¼ 0. This implies that u ¼ v ¼ 0, and so α<sup>1</sup> and α<sup>2</sup> must vanish. Hence, dt ¼ 0 which implies that, since the α<sup>i</sup> 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.
