4. Connection form associated to a coframe and transformation properties

Given the linearly independent one forms ω1;ω2, the first two of the structure equations uniquely determine the form ω12. The ω1;ω<sup>2</sup> is called the orthonormal coframe of the metric

$$ds^2 = a\_1^2 + a\_2^2,$$

and ω<sup>12</sup> is the connection form associated with it.

Theorem 4.1 Suppose that A > 0 is a function on M. Under the change of coframe

$$
\omega\_1^\* = A\omega\_1, \qquad \omega\_2^\* = A\omega\_2,\tag{21}
$$

the associated connection forms are related by

$$
\omega\_{12}^\* = \omega\_{12} + \
\
\
\omega\_{\text{12}} + \
\
\
d \log A. \tag{22}
$$

Proof: The structure equations for the transformed system are given as

$$d\omega\_1^\* = \omega\_{12}^\* \wedge \omega\_2^\*, \qquad d\omega\_2^\* = \omega\_1^\* \wedge \omega\_{12}^\*.$$

Using Eq. (21) to replace the ω� <sup>i</sup> in these, we obtain

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

Proof: Differentiating both sides of Lemma 3.3 gives

Equating the coefficients of t to zero gives the result (20).

and ω<sup>12</sup> is the connection form associated with it.

the associated connection forms are related by

that this equation is completely integrable.

curvatures fixed if and only if

26 Manifolds - Current Research Areas

theorem originally due to Bonnet.

angle.

metric

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

Theorem 3.2 A surface M admits a nontrivial isometric deformation that keeps the principal

dt∧α<sup>1</sup> þ tdα1−dα<sup>2</sup> ¼ ðtα1−α2Þ∧α<sup>1</sup> þ tdα1−dα<sup>2</sup> ¼ 0:

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 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

4. Connection form associated to a coframe and transformation properties

Given the linearly independent one forms ω1;ω2, the first two of the structure equations uniquely determine the form ω12. The ω1;ω<sup>2</sup> is called the orthonormal coframe of the

> <sup>1</sup> <sup>þ</sup> <sup>ω</sup><sup>2</sup> 2,

> > <sup>2</sup> ¼ Aω2, (21)

<sup>12</sup> ¼ ω<sup>12</sup> þ � dlogA: (22)

ds<sup>2</sup> <sup>¼</sup> <sup>ω</sup><sup>2</sup>

<sup>1</sup> ¼ Aω1; ω�

Theorem 4.1 Suppose that A > 0 is a function on M. Under the change of coframe

ω�

ω�

Proof: The structure equations for the transformed system are given as

dα<sup>1</sup> ¼ 0; dα<sup>2</sup> ¼ α1∧α2: (20)

$$d\log A \wedge \omega\_1 + d\omega\_1 = \omega\_{12}^\* \wedge \omega\_2, \qquad d\log A \wedge \omega\_2 + d\omega\_2 = \omega\_1 \wedge \omega\_{12}^\*.$$

The ω<sup>i</sup> satisfy a similar system of structure equations, so replacing dω<sup>i</sup> here yields

$$(\omega\_{12}^\* \neg \omega\_{12}) \wedge \omega\_2 = d \log A \wedge \omega\_1, \qquad (\omega\_{12}^\* \neg \omega\_{12}) \wedge \omega\_1 = -d \log A \wedge \omega\_2.$$

Since the form ω<sup>i</sup> satisfies the equations �ω<sup>1</sup> ¼ ω<sup>2</sup> and �ω<sup>2</sup> ¼ −ω1, substituting these relations into the above equations and using Ω<sup>k</sup> ∧ ð�ΘkÞ ¼ Θ<sup>k</sup> ∧ ð�ΩkÞ, we obtain that in the form

$$
\omega\_1 \wedge \ast (\omega\_{12}^\* - \omega\_{12}) = -\omega\_1 \wedge d \log A, \qquad \omega\_2 \wedge \ast (\omega\_{12}^\* - \omega\_{12}) = -\omega\_2 \wedge d \log A.
$$

Cartan's lemma can be used to conclude from these that there exist functions f and g such that

$$\*(\omega\_{12}^\* - \omega\_{12}) = -d\log A - f\omega\_1, \qquad \*(\omega\_{12}^\* - \omega\_{12}) = -d\log A + g\omega\_2.$$

Finally, apply � to both sides and use �<sup>2</sup> <sup>¼</sup> <sup>−</sup>1 to obtain

$$
\omega\_{12}^\* - \omega\_{12} = \*d\log A + f\omega\_2, \qquad \omega\_{12}^\* - \omega\_{12} = \*d\log A + g\omega\_1.
$$

The forms ω<sup>i</sup> are linearly independent, so for these two equations to be compatible, it suffices to put <sup>f</sup> <sup>¼</sup> <sup>g</sup> <sup>¼</sup> 0, and the result follows. □

For the necessity in the Chern criterion, Theorem 3.2, no mention of the set V of critical points of H is needed. In fact, when H is constant, this criterion is met and the sufficiency also holds with τ constant. However, when H is not identically constant, we need to take the set V of critical points into account for the sufficiency. In this case, M−V is also an open, dense, and connected subset of M. On this subset J > 0 and the function A can be defined in terms of the functions u and v as

$$A = +\sqrt{\mu^2 + v^2} > 0.\tag{23}$$

To define more general transformations of the ωi, define the angle ψ as

$$u = A\cos\left(\psi\right), \qquad v = A\sin\left(\psi\right). \tag{24}$$

This angle, which is defined modulo 2π, is continuous only locally and could be discontinuous in a nonsimply connected region of M−V. With A and ψ related to u and v by Eq. (24), the forms ϑ<sup>i</sup> and α<sup>i</sup> can be written in terms of A and ψ as

$$\begin{aligned} \mathfrak{G}\_1 &= A(\cos(\psi)\,\,\omega\_1 + \sin(\psi)\,\omega\_2), & \mathfrak{G}\_2 &= A(-\sin(\psi)\,\,\omega\_1 + \cos(\psi)\,\,\omega\_2), \\ \mathfrak{a}\_1 &= A(\cos(\psi)\,\,\omega\_1 - \sin(\psi)\,\,\omega\_2), & \mathfrak{a}\_2 &= A(\sin(\psi)\,\,\omega\_1 + \cos(\psi)\,\,\omega\_2). \end{aligned} \tag{25}$$

The forms ωi, ϑi, α<sup>i</sup> define the same structure on M and we let ω12, ϑ12, α<sup>12</sup> be the connection forms associated to the coframes ω1;ω2; ϑ1;ϑ2; α1;α2. The next theorem is crucial for what follows.

## Theorem 4.2

$$
\Psi\_{12} = d\psi + \omega\_{12} + \*d\log A = 2d\psi + \alpha\_{12}.\tag{26}
$$

Proof: Each of the transformations which yield the ϑ<sup>i</sup> and α<sup>i</sup> in the form (25) can be thought of as a composition of the two transformations which occur in the Theorems 3.1 and 4.1. First apply the transformation ω<sup>i</sup> ! Aω<sup>i</sup> and τ ! −ψ with ω� <sup>i</sup> ! ϑ<sup>i</sup> in Eq. (15), we get the ϑ<sup>i</sup> equations in Eq. (25). Invoking Theorems 3.1 and 4.1 in turn, the first result is obtained

$$
\mathfrak{d}\_{12} = d\psi + \omega\_{12} + \*d\log A.
$$

The transformation to the α<sup>i</sup> is exactly similar except that τ ! ψ, hence

$$\alpha\_{12} = -d\psi + \omega\_{12} + \,\,\ast \,\, d\log A.$$

This implies �dlog A ¼ α<sup>12</sup> þ dψ−ω12. When replaced in the first equation of (26), the second equation appears. Note that from Theorem 3.2, α<sup>12</sup> ¼ α2, so the second equation can be given as ϑ<sup>12</sup> ¼ 2dψ þ α2.

Differentiating the second equation in Eq. (14) and using dα<sup>1</sup> ¼ 0, it follows that

$$d \, \*\, a\_{12} = 0.\tag{27}$$

Lemma 4.1 The angle ψ is a harmonic function d � dψ ¼ 0 and moreover, d � ϑ<sup>12</sup> ¼ 0.

Proof: From Theorem 4.2, it follows by applying � through Eq. (26) that

$$
\omega \* \mathfrak{G}\_{12} = \*d\omega\_{12} + \*d\psi \text{--} d\log A = \mathfrak{D} \*d\psi \text{--} \alpha\_1. \tag{28}
$$

Exterior differentiation of this equation using d � ω<sup>12</sup> ¼ 0 immediately gives

$$d \ast d\psi = 0.$$

This states that ψ is a harmonic function. Equation (28) also implies that d � ϑ<sup>12</sup> ¼ 0.
