Surfaces: from generalised Weierstrass representation to Cartan moving frame

Claude Pierre Massé

Laboratoire de Physique Fondamentale

18 September 2016

The generalised Weierstraß representation of a surface immersed in the three dimensional Euclidean space is derived, and then put in the moving frame form. Various topics around surface theory are discussed.

Version history
• [v1] 18 September 2016
• [v2] 11 February 2017
• [v3] 21 April 2018

Introduction

The Enneper-Weierstraß representation allows to represent a minimal surface embedded in $\Bbb R^3$ by a holomorphic function and a holomorphic $1$-form. Recently, Konopelchenko [1] proposed a similar representation with two functions given by a solution of a two-dimensional Dirac equation. What is developed in this article isn’t really new, but is presented in a more transparent and systematic way. In §1, we first derive the generalized Weierstraß representation starting from the vector representation of a spinor. Everything is expressed in an arbitrary, fixed reference frame, then in §2, the representation is recast along the lines of the Cartan’s moving frame. The application in the field of integrable non-linear partial differential equations and their flat connection representation is briefly mentioned, and the mathematical expressions of the traditional data of a surface are spelled out each time. Finally in the last section §3, other points of view are touched upon.

§1. The generalized Weierstraß representation

The Dirac equation can be written in a Euclidean two-dimensional space using the Pauli matrices: $$\DeclareMathOperator{\Tr}{Tr}\DeclareMathOperator{\d}{d\!}\DeclareMathOperator{\e}{e} \sigma_1=\pmatrix{0& 1\cr 1& 0},\quad\sigma_2=\pmatrix{0& -i\cr i& 0},\quad\sigma_3=\pmatrix{1& 0\cr 0& -1}\tag{1.1}$$ as follow: $$\lbrace i\sigma_1\partial_\tau+i\sigma_2\partial_\xi+m\rbrace\psi=0.\tag{1.2}$$ Here the mass parameter $m$ is generalized to a real function of $\tau$ and $\xi.$ We shall work with complex numbers for compactness, and denote the complex conjugate by an overbar. We first define the differential operators $$\partial=\frac12(\partial_\tau-i\partial_\xi),\quad\bar\partial=\frac12(\partial_\tau+i\partial_\xi),\tag{1.3}$$ or using the complex variable: $$\zeta=\tau+i\xi,\tag{1.4}$$ $$\partial=\partial_\zeta,\quad\bar\partial=\partial_{\bar\zeta}.\tag{1.5}$$ Writing the wave function as: $$\psi=\binom\varphi\chi,\tag{1.6}$$ the Dirac equation (1.2) is now decomposed into a system of two simple and handy equations: \left\{\eqalign{ \bar\partial\varphi& =\frac i2m\chi\\ \partial\chi& =\frac i2m\varphi. }\right.\tag{1.7a,b} If $m$ vanishes identically, $\varphi$ is a holomorphic and $\chi$ an antiholomorphic function.

Defining the real numbers: $$\rho=\e^u:=\bar\varphi\varphi+\bar\chi\chi,\tag{1.8}$$ these differential constraints are deduced from the Dirac equation (1.7): $$\bar\varphi\partial\bar\varphi-\bar\chi\bar\partial\bar\chi=0=\chi\partial\chi-\varphi\bar\partial\varphi,\tag{1.9a}$$ $$\bar\varphi\partial\chi+\varphi\bar\partial\bar\chi=0=\chi\partial\bar\varphi+\bar\chi\bar\partial\varphi,\tag{1.9b}$$ $$\varphi\partial\bar\varphi+\bar\chi\partial\chi=0=\bar\varphi\bar\partial\varphi+\chi\bar\partial\bar\chi,\tag{1.9c}$$ $$\bar\varphi\partial\chi-\chi\partial\bar\varphi=\frac i2\rho m=\bar\chi\bar\partial\varphi-\varphi\bar\partial\bar\chi,\tag{1.9d}$$ $$\bar\varphi\partial\varphi+\chi\partial\bar\chi=\rho\partial u,\quad \rho\bar\partial u=\varphi\bar\partial\bar\varphi+\bar\chi\bar\partial\chi.\tag{1.9e}$$ We complete them by adding the defining relation: $$\bar\chi\partial\varphi-\varphi\partial\bar\chi=:\frac i2\rho Q,\quad \frac i2\rho\bar Q:=\bar\varphi\bar\partial\chi-\chi\bar\partial\bar\varphi.\tag{1.9f}$$

Remark that if $Q(\tau,~\xi)$ is given and $\rho$ is constant, $\psi$ is fully determined if it is known at a given point. The function $Q$ is further invariant under a rotation in the $(\tau,~\xi)$ plane, it is a scalar, but it isn’t arbitrary as it will be seen later. The related quantity $J$ used by Bracken [3] (eq. (3.7)) is $J=\frac12\rho Q.$ Taking the derivative of the defining relation (1.9f) and using the Dirac equation, we directly get $$\rho^{-1}\bar\partial(\rho Q)=\rho\partial(\rho^{-1}m).\tag{1.10}$$ The quantity $Q$ could be called the mirror mass, and we have the mirror Dirac equation \left\{\eqalign{ \partial\varphi& =\frac i2Q\chi+\partial u~\varphi\\ \bar\partial\chi& =\frac i2\bar Q\varphi+\bar\partial u~\chi. }\right.\tag{1.11a,b} It is easily verifiable that the differential constraints (1.9) are satisfied.

We now focus on the spinor. We use the well known vector representation. First, the spinor is uniquely extended as follow: $$\Phi=\pmatrix{\varphi& -\bar\chi\cr \chi& \bar\varphi}.\tag{1.12}$$ It is a unitary matrix multiplied by the factor $\sqrt\rho.$ All of the last differential constraints (1.9c-f) are summed up in the single matrix equation $$\d\Phi\Phi^\dagger=\rho\pmatrix{\partial u\d\zeta& \frac i2(m\d\bar\zeta+Q\d\zeta)\\ \frac i2(m\d\zeta+\bar Q\d\bar\zeta)& \bar\partial u\d\bar\zeta}.\tag{1.13}$$ A matrix such as $\Phi$ represents a rotation times a scaling in $\Bbb R^3$ as follow: $$\boldsymbol V=v^i\sigma_i\to\boldsymbol V'=\Phi\boldsymbol V\Phi^\dagger.\tag{1.14}$$ The matrix $\Phi$ can thus be represented by a multiple of an orthogonal $3\times3$ real matrix, whose elements are easily found by using the inner product in the space of $\boldsymbol V$: $$\langle\boldsymbol V_1,~\boldsymbol V_2\rangle=\frac12\Tr[\boldsymbol V_1\boldsymbol V_2],\tag{1.15}$$ then $$e^j_i=\frac12\Tr[\sigma_i\Phi\sigma_j\Phi^\dagger],\tag{1.16}$$ and after some straightforward calculations: $$e={\small\pmatrix{\boldsymbol e_+:={\sqrt2\over2}(\boldsymbol e_1-i\boldsymbol e_2)\\ \boldsymbol e_-:={\sqrt2\over2}(\boldsymbol e_1+i\boldsymbol e_2)\\ \boldsymbol e_3}} ={\small\pmatrix{{\sqrt2\over2}(\varphi\varphi-\bar\chi\bar\chi)& -i{\sqrt2\over2}(\varphi\varphi+\bar\chi\bar\chi)& \sqrt2~\varphi\bar\chi\\ {\sqrt2\over2}(\bar\varphi\bar\varphi-\chi\chi)& i{\sqrt2\over2}(\bar\varphi\bar\varphi+\chi\chi)& \sqrt2~\bar\varphi\chi\\ -(\varphi\chi+\bar\varphi\bar\chi)& i(\varphi\chi-\bar\varphi\bar\chi)& \bar\varphi\varphi-\bar\chi\chi}.\tag{1.17}}$$

According to the properties of the inner product, we can write $$\boldsymbol E_i:=e^j_i\sigma_j=\Phi^\dagger\sigma_i\Phi,\tag{1.18}$$ with which we calculate $$\d\boldsymbol e_i\cdot\boldsymbol e_j=\langle\d\boldsymbol E_i,~\boldsymbol E_j\rangle =\frac12\Tr[\d\,(\Phi^\dagger\sigma_i\Phi)\Phi^\dagger\sigma_j\Phi].\tag{1.19}$$ Differentiating and using $\Phi\Phi^\dagger=\rho I,$ this gives $$\d\boldsymbol e_i\cdot\boldsymbol e_j= \frac12\rho\Tr[\d\Phi\Phi^\dagger\sigma_j\sigma_i+(\d\Phi\Phi^\dagger\sigma_j\sigma_i)^\dagger]= \frac12\rho~{\frak Re}\{\Tr[\d\Phi\Phi^\dagger\sigma_j\sigma_i]\},\tag{1.20}$$ from which we deduce the vector representation of the differential constraints (1.13) $$\d ee^\dagger=\rho^2{\small\pmatrix{ 2\partial u\d\zeta& 0& -i{\sqrt2\over2}(m\d\bar\zeta+Q\d\zeta)\\ 0& 2\bar\partial u\d\bar\zeta& i{\sqrt2\over2}(m\d\zeta+\bar Q\d\bar\zeta)\\ -i{\sqrt2\over2}(m\d\zeta+\bar Q\d\bar\zeta)& i{\sqrt2\over2}(m\d\bar\zeta+Q\d\zeta)& \partial u\d\zeta+\bar\partial u\d\bar\zeta }}.\tag{1.21}$$

Remark: There is a two-ways fast lane for switching between both representations. If the generators of $SU(2)$ associated with $\Phi$ are $\frac i2\sigma_i,$ the corresponding generators of $SO(3),$ whose Lie algebra is the same, associated with $e$ are $$\Sigma_1=\frac{\sqrt2}2{\small\pmatrix{0& 0& -i\\0& 0& i\\-i& i& 0}},~ \Sigma_2=\frac{\sqrt2}2{\small\pmatrix{0& 0& -1\\0& 0& -1\\1& 1& 0}},~ \Sigma_3={\small\pmatrix{i& 0& 0\\0& -i& 0\\0& 0& 0}.\tag{1.22}}$$ Similarly, the generator of the scaling is $\frac12I$ for he spinor representation, and $I$ for the vector representation, where $I$ is the unit matrix of the appropriate size. The matrix constraints belong to the respective Lie algebra. One passes from one to the other just by replacing the basis of the Lie algebra representation. Denoting this abstract basis by $S_i,$ $i$ running from $0$ to $3,$ the constraints (1.13), (1.21) read $$\d {\cal II}^\dagger=\e^{2uS_0}\{\d u~S_0 -i(\partial u\d\zeta-\bar\partial u\d\bar\zeta)S_3+\\ +\frac12[(m+Q)\d\zeta+(m+\bar Q)\d\bar\zeta]S_1 -\frac i2[(m-Q)\d\zeta-(m-\bar Q)\d\bar\zeta]S_2\},\tag{1.23}$$ where ${\cal I}$ stands either for $\Phi$ or $e.$ The same works as well for $\Phi$ and $e$ using the exponential map of the Lie algebra, from which the differential constraints can be derived. It is still simpler using the step generators $S_\pm=S_1\pm iS_2$ and $\tilde S_\pm=S_0\pm iS_3,$ but I shall not spoil the fun of the reader, who may want to prove the claim too.

The surface is constructed by using the family of orthogonal frames parameterised by $(\zeta,~\bar\zeta)$ that we get through the spinor function. Its normal is taken to be $\boldsymbol e_3$ so that it is a conformal immersion in $\Bbb R^3.$ By mere inspection of the expression of $\d ee^\dagger$ (1.21), we read the simple equation $$\partial\boldsymbol e_-+\bar\partial\boldsymbol e_+=0.\tag{1.24}$$ Each component of $\boldsymbol e_-$ and $\boldsymbol e_+$ is then a conserved current, the third one is just the Dirac (electromagnetic) current, explicitly: $$\frac{\sqrt2}2\pmatrix{\varphi\varphi-\bar\chi\bar\chi\\ \bar\varphi\bar\varphi-\chi\chi},\quad i\frac{\sqrt2}2\pmatrix{-\varphi\varphi-\bar\chi\bar\chi\\ \phantom-\bar\varphi\bar\varphi+\chi\chi},\quad \sqrt2\pmatrix{\varphi\bar\chi\\ \bar\varphi\chi}.\tag{1.25}$$ Then the $\Bbb R^3$-valued differential form $$\d\boldsymbol r=i\frac{\sqrt2}2\boldsymbol e_+\d\zeta-i\frac{\sqrt2}2\boldsymbol e_-\d\bar\zeta\tag{1.26}$$ is closed, and in addition it is normal to $\boldsymbol e_3,$ and real because $\boldsymbol e_-=\bar{\boldsymbol e}_+.$ Moreover, it is exact in virtue of the Poincaré’s lemma since every loop in $\Bbb C$ can be shrunk to a point. Therefore, integrating it along a path $\Gamma$ in the $(\zeta,~\bar\zeta)$ plane from a fixed point $\zeta_0,$ we arrive at the wanted equation of the surface: \eqalignno{ x+iy& =i\int_\Gamma(\varphi^2\d\zeta'+\chi^2\d\bar\zeta')+x_0+iy_0& (1.27a)\\ x-iy& =i\int_\Gamma(-\bar\chi^2\d\zeta'-\bar\varphi^2\d\bar\zeta')+x_0-iy_0& (1.27b)\\ z& =i\int_\Gamma(\varphi\bar\chi\d\zeta'-\bar\varphi\chi\d\bar\zeta')+z_0& (1.27c) } for the point $\boldsymbol r$ of coordinates $(x,y,z)$ in $\Bbb R^3.$ That is the same as in [1] (equation 2) up to some cosmetic rearrangements.

From this derivation, it is obvious that the Gauß map is directly given by: $$\boldsymbol\phi=\boldsymbol e_3/\rho.\tag{1.28}$$ The north pole corresponds to $\chi=0,$ and the south pole to $\varphi=0.$ To go further, let us remark that in the vector representation, we use row vectors while the spinor is a column. If we consider the row spinor $(\varphi,-\bar\chi)$ and the associated stereographic projection, or fundamental field: $$w=-\bar\chi/\varphi,\tag{1.29}$$ then, dividing the numerator and the denominator by $\bar\varphi\varphi$ in the expression of $\boldsymbol\phi$ (1.28), we obtain $$\boldsymbol\phi=(w+\bar w,~i(w-\bar w),~1-\bar ww)/(1+\bar ww),\tag{1.30}$$ that is a function of $w$ alone. In other words, $w$ is also the stereographic projection of the Gauß map from the south pole. Moreover, for a minimal surface ($m=0,$) it is readily shown that $w$ is an analytic function of $\zeta,$ as is known for a long time. That generalises for any non-minimal surface with $w$ being any function.

We now have all what is necessary to get the traditional data of a surface. Since $\boldsymbol e_\pm^2=0$ and $\boldsymbol e_+\cdot\boldsymbol e_-=\rho^2,$ the first fundamental form is $${\rm I}:=\d\boldsymbol r\cdot\d\boldsymbol r=\rho^2\d\zeta~\d\bar\zeta\tag{1.31}$$ (the coordinates $\tau,~\xi$ are therefore isothermal.) The second fundamental form can be directly calculated too: $${\rm II}:=-\d~(\rho^{-1}\boldsymbol e_3)\cdot\d\boldsymbol r= -\rho^{-1}\d\boldsymbol e_3\cdot\d\boldsymbol r= -\frac12\rho(Q\d\zeta\d\zeta+2m\d\zeta\d\bar\zeta+\bar Q\d\bar\zeta\d\bar\zeta).\tag{1.32}$$ The term $Q\d\zeta\d\zeta$ is what is called the Hopf differential. Finally the usual formulas gives the mean curvature $H$ and the gauß curvature $K$ : $$H=\frac12\Tr[{\rm II}\cdot{\rm I}^{-1}]=-\rho^{-1}m,\tag{1.33}$$ $$K=\det({\rm II}\cdot{\rm I}^{-1})=\rho^{-2}(m^2-Q\bar Q)=H^2-\rho^{-2}Q\bar Q.\tag{1.34}$$ Konopelchenko [1] (equation 4) gives a different expression: $$K=-4\rho^{-2}\partial\bar\partial u,\tag{1.35}$$ which is known as the Gauß-Riemann curvature. But Ferapontov and Grundland [2] (equation 2.6) proved that they are equivalent, in accordance with Gauß’ Theorema Egregium. Then the modulus of $Q$ is a function of $\rho$ and $m$: $$Q\bar Q=m^2+4\partial\bar\partial u.\tag{1.36}$$

§2. The Cartan moving frame

So far, we implicitly used a fixed referential frame in $\Bbb R^3.$ Indeed, by varying this frame, with the same solution of the Dirac equation we get a whole set of surfaces that are deduced from each other by a rigid motion. This way has been deemed awkward by Cartan, so he developed the powerful method of the moving frame [4]. To an infinitesimal displacement in the $(\zeta,~\bar\zeta)$ plane is now associated an infinitesimal change of the origin and of the basis vectors, but this time expressed with respect to the moving frame itself. By using differential $1$-forms (also called Pfaffian forms,) this is written down as \left\lbrace\eqalign{ \d\boldsymbol r& =\omega^j\boldsymbol e_j\\ \d\boldsymbol e_i& =\omega^j_i\boldsymbol e_j }\right.,\quad i,j=+,-,3,\tag{2.1a,b} which is called the moving frame equations. The vectors $\boldsymbol e_i$ satisfy the orthonormality conditions $$\boldsymbol e_\pm^2=0,\quad\boldsymbol e_+\cdot\boldsymbol e_-=\rho^2, \quad\boldsymbol e_\pm\cdot\boldsymbol e_3=0,\quad\boldsymbol e_3^2=\rho^2.\tag{2.2}$$ Differentiating them, and using the moving frame equations (2.1), we get relations among the differential forms, e.g.: $$\d\boldsymbol e_+^2=0\Rightarrow 2\boldsymbol e_+\cdot\d\boldsymbol e_+=0\Rightarrow \omega^-_+=0.\tag{2.3}$$ By a similar method, we collect the following relations: $$\omega^+_-=\omega^-_+=0,\tag{2.4a}$$ $$\omega^-_3+~\omega^3_+=\omega^+_3+~\omega^3_-=0.\tag{2.4b}$$ $$\omega^-_-+~\omega^+_+=2\d u,\tag{2.4c}$$ $$\omega^3_3=\d u,\tag{2.4d}$$ As $\bar{\boldsymbol e}_+=\boldsymbol e_-$ and $\bar{\boldsymbol e}_3=\boldsymbol e_3,$ with the complex conjugates of the same equations we moreover find \eqalignno{ \bar\omega^3_+& =\omega^3_-,&(2.5a)\\ \bar\omega^+_+& =\omega^-_-.&(2.5b) } Then assuming the moving frame equations are integrable, by differentiating them and substituting $\d\boldsymbol e_i$ (2.1b) wherever possible, we get compatibility equations, called the structure equations by Cartan: \left\{\eqalign{ \d\omega^i& =\omega^k\wedge\omega^i_k\\ \d\omega^i_j& =\omega^k_j\wedge\omega^i_k. }\right.\tag{2.6a,b} This condition is the equivalent, expressed in the moving frame language, of the requirement that the form $\d\boldsymbol r$ be exact.

From the expression of $\d\boldsymbol r$ (1.26) we immediately get $$\omega^+=i\frac{\sqrt2}2d\zeta,\quad\omega^-=-i\frac{\sqrt2}2\d\bar\zeta,\quad\omega^3=0,\tag{2.7}$$ and $$\bar\omega^+=\omega^-.\tag{2.8}$$

The expression of the remaining $1$-forms are easily obtained by writing the second moving frame equation (2.1b) with a matrix: $$\d e=\Omega e,\tag{2.9}$$ where $$\Omega=\pmatrix{ \omega^+_+& \omega^-_+& \omega^3_+\\ \omega^+_-& \omega^-_-& \omega^3_-\\ \omega^+_3& \omega^-_3& \omega^3_3\\ }.\tag{2.10}$$ Multiplying by $e^\dagger$ from the right, there results $$\Omega=\rho^{-2}\d ee^\dagger\tag{2.11}$$ which is already known, it is our differential constraints (1.21) with an additional factor, the preceding relations (2.4), (2.5) are incidentally satisfied. Further writing $$\Theta=(\omega^+,~\omega^-,~0),\tag{2.12}$$ the structure equations (2.6) become \left\{\eqalign{ \d\Theta& =\Theta\wedge\Omega\\ \d\Omega& =\Omega\wedge\Omega. }\right.\tag{2.13a,b} If we decompose $\Omega$ as follow: $$\Omega={\cal Z}\d\zeta+\tilde{\cal Z}\d\bar\zeta,\tag{2.14}$$ with $${\cal Z}={\small\pmatrix{ 2\partial u& 0& -i{\sqrt2\over2}Q\\ 0& 0& i{\sqrt2\over2}m\\ -i{\sqrt2\over2}m& i{\sqrt2\over2}Q& \partial u }},\quad\tilde{\cal Z}={\small\pmatrix{ 0& 0& -i{\sqrt2\over2}m\\ 0& 2\bar\partial u& i{\sqrt2\over2}\bar Q\\ -i{\sqrt2\over2}\bar Q& i{\sqrt2\over2}m& \bar\partial u }},\tag{2.15}$$ the moving frame equations (2.1) are then $$\partial e={\cal Z}e,\quad\bar\partial e=\tilde{\cal Z}e,\tag{2.16}$$ which is known in the theory of surfaces as the Gauß-Weingarten equations. The second structure equation (2.6b) reads $$\bar\partial{\cal Z}-\partial\tilde{\cal Z}+[{\cal Z},~\tilde{\cal Z}]=0,\tag{2.17}$$ also known as the Gauß-Codazzi-Ricci equation. Many integrable non-linear partial differential equations are expressed within this formalism. The last equation (2.17) is then called the zero curvature condition, and is the non-linear equation represented as a compatibility condition of a linear system.

In the spinor representation, the corresponding matrices are readily obtained by switching representation (1.23), they are $$Z=\frac12\pmatrix{ 2\partial u& iQ\\ im& 0},\quad \tilde Z=\frac12\pmatrix{ 0& im\\ i\bar Q& 2\bar\partial u },\tag{2.18}$$ and the equations (2.16), (2.17) become $$\partial\Phi=Z\Phi,\quad\bar\partial\Phi=\tilde Z\Phi,\tag{2.19}$$ $$\bar\partial Z-\partial\tilde Z+[Z,\tilde Z]=0.\tag{2.20}$$

Spelling out the structure equations (2.6), we easily derive the already known equations (1.10) and (1.36) again. In addition, as $\omega^3=0,$ from the structure equation for $\d\omega^3$ we have $$0=\omega^+\wedge\omega^3_++\omega^-\wedge\omega^3_-.\tag{2.21}$$ According to the Cartan’s lemma, $\omega^3_+$ and $\omega^3_-$ are then equal to a linear combination of $\omega^+$ and $\omega^-,$ like \eqalign{ \omega^3_+& =h_{++}\omega^+& +h_{+-}\omega^-\\ \omega^3_-& =h_{-+}\omega^+& +h_{--}\omega^-, }\tag{2.22a,b} and $$h_{-+}=h_{+-},\tag{2.23}$$ as can be seen by substituting $\omega^3_+$ and $\omega^3_-$ in the equation (2.21). Finally by identification we get $$h_{++}=-Q,\quad h_{--}=-\bar Q,\tag{2.24a}$$ $$h_{+-}=h_{-+}=m,\tag{2.24b}$$ or $$h=\pmatrix{ -Q& m\\ m& -\bar Q }.\tag{2.25}$$

Now using this matrix together with the moving frame equations (2.1), we are able to express the data of the surface in term of the Pfaffian forms: $${\rm I}=2\rho^2\omega^+\omega^-,\tag{2.26}$$ $${\rm II}=-\rho(\omega^+\omega^3_++\omega^-\omega^3_-) =-\rho(h_{++}\omega^+\omega^++(h_{+-}+h_{-+})\omega^+\omega^-+h_{--}\omega^-\omega^-).\tag{2.27}$$ The surface element is $$\d S=i\rho^2\omega^+\wedge\omega^-,\tag{2.28}$$ and the corresponding surface element on the Gauß map $$\d\sigma=i\omega^3_-\wedge\omega^3_+=i(h_{+-}h_{-+}-h_{++}h_{--})\omega^+\wedge\omega^-.\tag{2.29}$$ The total curvature is the ratio of these two surfaces: $$K={\d\sigma\over\d S}=\rho^{-2}(h_{+-}h_{-+}-h_{++}h_{--}),\tag{2.30}$$ and the mean curvature is \eqalign{ H& =\frac i2\rho{\omega^-\wedge\omega^3_--\omega^+\wedge\omega^3_+\over\d S}\\ & =-\frac i2\rho{(h_{+-}+h_{-+})\omega^+\wedge\omega^-\over \d S}=-\frac12\rho^{-1}(h_{+-}+h_{-+}). }\tag{2.31} The results are the sames as in the end of the previous section (1.31-34), as expected.

§3. Other

1 - Connection

The differential form $\Omega$ (2.11) is a connection, and actually an affine connection of $\Bbb R^3$ that is flat on the surface, which is the meaning of the structure equations. We have thus the stunning result that the (two-dimensional) Dirac equation is but a way of writing an affine connection. Its restriction to the surface is $$\Omega_{/2}=2\pmatrix{ \partial u\d\zeta& 0\\ 0& \bar\partial u\d\bar\zeta }\tag{3.1}$$ and its intrinsic scalar curvature is precisely the Riemann-Gauß curvature of the surface. The restriction of the structure equations doesn’t necessarily hold, and indeed we have: $$\d\omega_+^+=\omega_+^+\wedge\omega_+^++\omega_+^-\wedge\omega_-^++\omega_+^3\wedge\omega_3^+=\omega_+^3\wedge\omega_3^+.\tag{3.2}$$ But $\omega_+^3$ isn’t defined on the surface, thus the term $\omega_+^3\wedge\omega_3^+=\omega^3_-\wedge\omega^3_+$ stands for the affine curvature form, whose explicit expression taken in three dimensions, that is extrinsically, is the Gauß curvature. That’s still another way of looking at the Theorema Egregium. There is no torsion from the sheer fact that $\omega^3=0,$ and thus it is indeed the Levi-Civita connection.

2 - Symmetries

For investigating the different global symmetries, let us first remark that if we make the substitution $$\matrix{ \varphi& \to& -\bar\chi\\ \chi& \to& \bar\varphi, }\tag{3.3}$$ which is in fact the charge conjugaison, the Dirac equation (1.7) still holds. That is, we get the same solutions if we put $\Phi$ instead of $\psi$ in equation (1.2). Then, $\Phi$ multiplied from the right by any constant non singular matrix is a solution of the equation if $\Phi$ is so. Thus, the full symmetry group is $GL(2,\Bbb C).$ The substitution above belongs to this group, which makes charge conjugaison a continuous transformation (i.e. continuously connected to identity.) The surface is determined by the functions $\rho$ and $Q$ alone, its transformation then depends only on the transformation properties of these functions. For example, the substitution (3.3) leaves them invariant, so the surface doesn’t change. More generally, the function $\rho$ is invariant under the unitary subgroup $SU(2),$ in this case only the function $Q$ is to be examined. As both curvatures remain invariant under this subgroup since they are independent of $Q$, the surfaces are identical up to a rigid motion. Actually, in virtue of equation (1.36) only the global phase of $Q$ varies.