General Spinor Representation of Submanifolds and Clifford Algebras – Framework

Claude Pierre Massé

Laboratoire de Physique Fondamentale, FRANCE

© Claude Pierre Massé 2023

9 May 2023

The Clifford Algebra language for describing the immersion of a (pseudo-)Riemannian manifold of dimension $p$ in $\Bbb R^n,$ $n\gt p$ is developed and shown to be the adequate one. Any signature of these spaces is possible by choosing the corresponding signature of the Clifford algebra, thus the spinor representation is generalised to any signature and extended to any dimension. Then is investigated whether the Dirac equation can be introduced too. Some results are given, especially for a hypersurface the equivalence with the Dirac equation is proven, and for the other cases a weaker solution is found. The extended Gauß map and its stereographic projection is also worked out in the Clifford language, and it is shown how it could be lifted. Finally all the immersion formulæ known so far are derived. The application of this framework to the multidimensional inverse scattering transform is discussed in the conclusion.

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Introduction

The embedding or immersion of a surface in a flat Euclidean space of at least $3$ dimensions, both locally (in the small) and globally (in the large,) is a well understood topic of geometry. It has been already studied in the classical period where the most important concepts and nonlinear equations have been introduced. Recently the old Weierstraß-Enneper induction formula has been refurbished and generalised by using the more modern Dirac equation [K96]. This being, not much progress have been done in extending the immersion to manifolds of higher dimensions, a prerequisite to address realistic problems in physics. As will rapidly be evident, this is due to formidable algebraic barriers, that need to be tackled with more powerful calculational tools. Clifford algebras provide such tools, as shown here.

Konopelchenko’s formulation is basically analytic, and uses two complex variables. The idea is that instead of an element of the minimal left ideal, the spinor satisfying the Dirac equation should be taken in the sense of Hestenes’ operator spinor as an element of the special Clifford group associated to the ambiant space [BLR’17, V’19]. For the simplest case of a surface in $\Bbb R^3,$ the special Clifford group $Γ^0_{3,0}$ is isomorphic to $\Bbb R^*⊗{\rm SU}(2),$ or to the set of non-singular matrices of the form $$\pmatrix{φ& -\barχ\\χ& \phantom-\barφ},\quadφ,χ\in\Bbb C.$$ Taking as the matrix representation of the Clifford generators $$E_1=i\pmatrix{0& 1\\1& 0},\quad E_2=i\pmatrix{0& -i\\i& \phantom-0},\quad E=\pmatrix{1& 0\\0& 1},$$ it is rewriten as $$Ψ=\pmatrix{φ'+iφ''& -χ'+iχ''\\χ'+iχ''& \phantom-φ'-iφ''} =φ'E+χ''E_1-χ'E_2-φ''E_{12}=Ψ^IE_I\in{\rm C}\ell_{2,0},\quadΨ^I\in\Bbb R.$$ It is but a compacted version, that we expand by using the isomorphism $${\rm C}\ell_{n-1,0}\hookrightarrow{\rm C}\ell_{n,0}^0,$$ generated by $$E_i\mapsto E_{ni},$$ where ${\rm C}\ell^0$ is the even subalgebra of ${\rm C}\ell.$ Then $Ψ$ really is an element of $Γ^0_{3,0}$ representing a mere rotation.

So similarly as a surface is immersed in an Euclidean space of greater dimension, a Clifford algebra is immersed in another algebra of higher order. It is a local homomorphism of Clifford algebras realising a Clifford bundle. As all the developments will be local, issues like the existence of a spin structure do not arise, and the related, somewhat opaque formalism is bypassed.

Contrary to the traditional presentation order, it will be more direct to begin from the end. We shall also be very general, the known special cases are given in the last section only as a result. We begin by a compendium about the Clifford algebras in order to make available all that is necessary to use them here §1. Then Cartan’s method of the moving frame is spelled out and adapted to the Clifford algebras §2. The most important part is the thorough investigation of the use of the Dirac equation for the representation of a submanifold by an immersion in an Euclidean space §3. A essential concept in these problems is the Gauß map, that is worked out and extended in §4. The last section is devoted to recover the concrete formula for particular cases that are found in the literature §5. Finally we conclude by an outlook of the possible application and open problems, especially for nonlinear partial differential equations.

§1. Clifford algebras

The strenuous effort of Hamilton to multiply vectors eventually led to one of the most imitated mathematical structure : from Clifford original proposition, passing by the multenions and multivectors, to Dirac matrices and Eddington’s $E$-numbers, until fully exploited by Hestenes. True to the dream of Hamilton, the Clifford algebras provides all the tools to handle geometrical problems. Vectors represent the additive property of geometry, they see it as a juxtaposition of parallelipipeds (or parallelitopes,) always in the same dimension. Clifford algebras in contrast address the multiplicative properties. Their basic object is the sphere, and by combining, or more precisely multiplying them, get spheres of higher dimensions. That is why they are particularly well adapted to representations of rotation groups, be they real, complex, or symplectic, and then to geometry in the philosophy of Klein’s Erlangen program.

We won’t need the full abstract theory of Clifford algebras here. It is presented in many texts, the author recommends [L’01] for a pedagogical account and more details. Only what is necessary in this memoir is presented. We consider real Clifford algebras associated to a bilinear form in $\Bbb R^n.$ As the form can be made diagonal by a linear substitution of $\Bbb R^n,$ we restrict ourselves to diagonal bilinear forms with $+1$ or $-1$ as elements. For $n_+$ plus signs and $n_-$ minus signs, the Clifford algebra is denoted by ${\rm C}\ell_{n_+,n_-},$ and $(n_+,n_-)$ is called the signature. For the sake of notation simplicity, in such context we write $n$ for a dimension with an unspecified signature. We adopt the common convention in differential geometry that the basis elements $E_i$ of ${\rm C}\ell_{n_+,n_-}$ that correspond to a plus sign satisfy $E_iE_i=-E.$ Lowercase indices denote indices in $\Bbb R^n,$ while uppercase ones denote indices in ${\rm C}\ell_n,$ or equivalently, compound indices that contain several, one, or no single indices in $\Bbb R^n.$

A Clifford algebra like the ones we use is generated multiplicatively by a set of $n$ generating elements $E_1,\cdots,E_n,$ like $E_1E_2=E_{12}$ etc. The Clifford product in noted simply by juxtaposition, like matrices product. The algebra is closed by two types of relations : $$E_iE_j=-E_jE_i,\quad E_iE_i=\pm E.\tag{1.1}$$ The sign depends on the signature. A basis element of the full Clifford algebra can therefore have a maximum of $n$ indices. A generic element of a Clifford algebra, or Clifford number, can thus be written $$C=cE+c^iE_i+c^{ij}E_{ij}+\cdots\tag{1.2}$$

An involutive anti-automorphism called reverse is defined in the following way : $$(c^KE_K)^\Tilde=c^K\tilde E_K,\quad \skew5\tilde{\tilde E}_I=E_I,\quad \tilde E_i=E_i,\quad (E_IE_J)^\Tilde=\tilde E_J\tilde E_I\tag{1.3}$$ That simply means the reverse of $E_iE_j\cdots E_m$ is $E_m\cdots E_jE_i.$ Consequently we have $$\tilde E_I=(-)^{|I|(|I|-1)/2}E_I=(-)^{[|I|/2]}E_I\tag{1.4}$$ where $|I|$ is the number of distinct single indices in $I,$ called the grade. For the practical calculations it is useful to know that the sign change by reversion depends only on the grade modulo $4,$ and $$\tilde E=E,\quad\tilde E_i=E_i,\quad \tilde E_{ij}=-E_{ij},\quad\tilde E_{ijk}=-E_{ijk}.\tag{1.5}$$ More generally, the exchange of any two indices results in the change of sign.

In order to be able to make calculations that are valid for any signature, we define contravariant and covariant vector as usual: $$\eta_i=-E_iE_i,\quad E^i=\eta_iE_i,\tag{1.6}$$ where the indicator $\eta_i$ is the $i$-th diagonal element of the bilinear form, so that $$E^iE_i=E_iE^i=-E.\tag{1.7}$$ Similarly, compound indices look like $$E_I=E_{i\ph\ph\ell}^{\ph jk},\quad E^I=E^{\ell\ph\ph i}_{\ph kj} =(-)^{|I|(|I|-1)/2}\eta_i\eta_j\eta_k\eta_\ell E_I=:\eta_IE_I,\tag{1.8}$$ so that in the same way $$E_IE^I=E^IE_I=(-)^{|I|}E.\tag{1.9}$$ Beware the conventions for $\eta$ are different from usual for the Minkowski metric, in particular there is never summation in its index.

We use some short hand notations : $$(c^KE_K)^I=c^I,\quad \langle C\rangle_r=\sum_{|I|=r}(C)^IE_I\tag{1.10}$$ are the $I$ component and the $r$ grade of a Clifford number. $$E_P=E_{1\cdots p},\quad E_N=E_{1\cdots n}\tag{1.11}$$ are the unit pseudo scalars of ${\rm C}\ell_p$ and ${\rm C}\ell_n,$ while $$E_Q=E_{p+1\cdots n}.\tag{1.12}$$ For numerals we use the special greek letter $\iota,$ e.g. $$E_{\iota4}=E_{1234}\tag{1.13}$$

The $0$ grade $$\langle C\rangle:=\langle C\rangle_{_0}\tag{1.14}$$ has the properties of the trace. It is linear, and $$\langle AB\rangle =\langle (a^IE_I)(b^JE_J)\rangle =a^Ib^J\langle E_IE_J\rangle =b^Ja^I\langle E_JE_I\rangle =\langle BA\rangle,\tag{1.15}$$ $$\langle ABC\rangle=\langle CAB\rangle=\langle BCA\rangle,\tag{1.16}$$ while for all $r,$ $$\langle\tilde C\rangle_r=(-)^{r(r-1)/2}\langle C\rangle_r.\tag{1.17}$$

The most important structure of the Clifford algebras for us, and that generalises the spinor $\Phi,$ is the Clifford, or Lipschitz group $Γ_n.$ It is the subset of the elements that transforms a vector $V=v^iE_i$ into another vector $\boldsymbol V'=v'^iE_i$ as $$\boldsymbol V'=Ψ^{-1}VΨ.\tag{1.18}$$ The inverse comes first because we reserve the symbol $Ψ$ for the wave function. But that is only a matter of convention, the mathematics are the same by swapping $Ψ^{-1}\leftrightarrowΨ$ since it must be inversible from the definition. Though, at variance with the traditional twisted adjoint representation, we use a slightly different one, that has the advantage of being more faithful : $$Ψ(V)=\tildeΨ VΨ.\tag{1.19}$$ Since $\tildeΨΨ=:ρ E,$ this introduces an additional scaling. Finally the subgroup defined by $\tildeΨΨ=E$ is ${\rm Spin}(n),$ the double cover of ${\rm SO}(n).$ Actually, only the special Clifford group $Γ^0_n$ is used, that is, the elements that belong to the even sub algebra of the Clifford algebra. For this group, twisted and untwisted representations are identical. The reason is that the rotations have to be arbitrary close to the identity, which is necessary to describe a continuous surface. Similarly, a spinor formalism is used instead of a simpler vector one in order to represent non orientable surfaces. A local change of orientation can be performed in the ambient space by a proper rotation, while an improper one is required in the surface.

§2. Cartan moving frame

In all the following, greek indices are used for the index range $1,2,\cdots,p,$ Latin ones $a,b,\cdots, r,s,\cdots$ for $p+1,\cdots,n,$ while middle Latin ones $i,j,k,\ell$ are used for the full range $1,2,\cdots,n.$ Further or overriding conventions are specified where they are used.

2A. Classical formulation

The coordinates of a simply connected domain ${\cal D}$ of a manifold of dimension $p$ are noted by $\xi^μ,~μ=1,\cdots,p.$ ${\cal D}$ is immersed in $\Bbb R^n$ by the function $$\boldsymbol x:{\cal D}\to\Bbb R^n,\quad \xi=(\xi^1,\cdots,\xi^p)\mapsto\boldsymbol x(\xi)=(x^1,\cdots,x^n).\tag{2A.1}$$ To each point of the immersed space, that is for each value of $(\xi^1,\cdots,\xi^p),$ is attached an orthogonal frame $(\boldsymbol e_1,\cdots,\boldsymbol e_n)^t$ of $\Bbb R^n.$ In order to have a one-to-one correspondance with the Clifford group, the basis vectors are not normalised to unity, but they all have the same length, that is $$\boldsymbol e_i\cdot\boldsymbol e^j=ρ^2δ_i^j.\tag{2A.2}$$ The operation $\cdot$ is the associated bilinear form of $\Bbb R^n$ and $ρ$ is a real function over ${\cal D}.$ The first $p$ vectors are tangent to the immersed manifold, and the remaining $q=n-p$ ones are therefore normal to it. We formally write $n=p\oplus q.$

The frame field is differentiable, but instead of expressing the differentials in a fixed basis, they are expressed in the moving frame itself [C’01]. Differential $1$-forms $ω$ over ${\cal D}$ are defined through the structure equations of the space ($\d=\d\xi^μ∂_μ$) $$\left\{\eqalign{ \d\boldsymbol x& =ω^i\boldsymbol e_i,\\ \d\boldsymbol e_i& =ω_i^k\boldsymbol e_k, }\right.\tag{2A.3ab}$$ which, multiplying by $\boldsymbol e^j$ on the right, yields $$\left\{\eqalign{ ω^j& =ρ^{-2}\d\boldsymbol x\cdot\boldsymbol e^j,\\ ω_i^j& =ρ^{-2}\d\boldsymbol e_i\cdot\boldsymbol e^j. }\right.\tag{2A.4ab}$$ Here we have $$ω^a=0.\tag{2A.5}$$ Differentiating the orthonormality conditions (2A.2) we get $$ω_i^j+\eta_{ij}ω_j^i=2ρ^{-1}\dρ~δ_i^j\quad\text{(no sum.)}\tag{2A.6}$$

Then assuming the equations (2A.3ab) are integrable, that is, $\d\boldsymbol x$ is closed and the frame field defines a $p$-dimensional immersed manifold, by differentiating them and substituting the expression of $\d\boldsymbol e_i$ wherever possible, we get compatibility equations, called the structure equations of the group of motions by Cartan : $$\left\{\eqalign{ \dω^i& =ω^μ∧ω^i_μ\\ \dω^i_j& =ω^k_j∧ω^i_k. }\right.\tag{2A.7ab}$$

Differentiating further gives nothing new, unless the intrinsic and extrinsic part are kept separate. Since $ω^a=0,$ there is no torsion. The structure equations (2A.7ab) are written as $$\left\{\eqalign{ \dω^μ& =ω^\lambda∧ω^μ_\lambda\\ \dω^ν_μ& =ω^\lambda_μ∧ω^ν_\lambda+θ^ν_μ, }\right.\tag{2A.8ab}$$ in which $θ^ν_μ=ω^a_μ∧ω^ν_a$ is the curvature $2$-form. Through differentiation of these equations, substituting the differentials $\dω$ by their expression in the same equations, we get $$\left\{\eqalign{ ω^\lambda∧θ_\lambda^ν& =0\\ \dθ_μ^ν -ω_μ^\lambda∧θ_\lambda^ν+θ_μ^\lambda∧ω_\lambda^ν& =0={\frak d}θ_μ^ν. }\right.\tag{2A.9ab}$$ These are the Bianchi identities. Similar identities are obtained in the normal space by defining a second $2$-form $\tau$ : $$\dω_a^b=ω_a^c∧ω_c^b+\tau_a^b,\tag{2A.10}$$ that is $$\tau_a^b=ω_a^μ∧ω_μ^b,\tag{2A.11}$$ $$\d\tau_a^b-ω_a^c∧\tau_c^b+\tau_a^c∧ω_c^b=0={\frak d}\tau_a^b.\tag{2A.12}$$ The $2$-form $θ$ actually corresponds to the Riemann tensor, and $\tau$ is called the Riemannian or Gaußian torsion. (Here they are the same because the ambient space is flat.) The symmetries of these forms are $$θ^ν_μ+\eta_{μν}θ^μ_ν,\tag{2A.13}$$ $$\tau^b_a+\eta_{ab}\tau^a_b,\tag{2A.14}$$

The fundamental forms of the submanifold are given as usual by the differential forms $${\rm I}=\d\boldsymbol x\cdot\d\boldsymbol x =ρ^2\sum_μ\eta_μω^μω^μ,\tag{2A.I}$$ $${\rm II}_a=\d(ρ^{-1}\boldsymbol e_a)\cdot\d\boldsymbol x =ρ\sum_μ\eta_μω^μ_aω^μ,\tag{2A.II}$$ $${\rm III}_{ab}=\d(ρ^{-1}\boldsymbol e_a)\cdot\d(ρ^{-1}\boldsymbol e_b) =\sum_μω^μ_aω^μ_b.\tag{2A.III}$$

2B. Clifford formulation

Now we go over to the Clifford algebra. The vectors are replaced by the corresponding vectors of the algebra, that is : $$\left\{\eqalign{ \boldsymbol x& \to x^i E_i=\boldsymbol X\\ \boldsymbol e_i& \to e_i^j E_j=\boldsymbol E_i }\right.\tag{2B.1ab}$$ Moreover, the frame is expressed in the standard basis $(E_1,\cdots,E_n)$ through a function $Ψ(\xi)$ with values in the special Clifford group $Γ_n^0$ : $$\boldsymbol E_i=\tildeΨ E_iΨ.\tag{2B.2}$$ The system becomes $$\left\{\eqalign{ \d\boldsymbol X& =ω^μ\boldsymbol E_μ\\ \d\boldsymbol E_i& =ω_i^k\boldsymbol E_k }\right.\tag{2B.3ab}$$

In the last equation (2B.3b), there is still $ω$ that pertains to the vector representation. In fact it is the right invariant Maurer-Cartan form of the group $\Bbb R^*⊗{\rm SO}(n),$ and thus is an element of the Lie algebra ${\frak r\oplus so}(n).$ Its corresponding spinor representation is the right invariant Maurer-Cartan form of the special Clifford group, or $\Bbb R^*⊗{\rm Spin}(n)$ belonging to the Lie algebra ${\frak r\oplus spin}(n),$ that is $$Z=ρ^{-1}\dΨ\tildeΨ=\dΨΨ^{-1}.\tag{2B.4}$$ To work out this expression, let us write $$Ψ=ρ^{1/2}\hatΨ,\tag{2B.5}$$ where $\hatΨ\in {\rm Spin}(n).$ Inserting it in the expression $Z,$ we get $$ρ^{-1}\dΨ\tildeΨ=\tfrac12ρ^{-1}\dρ+\d\hatΨ\hatΨ^{-1}.\tag{2B.6}$$ Now $\d\hatΨ\hatΨ^{-1}\in{\frak spin}(n)=\span(E_i^{\ph j}/2)$ is a bivector. We denote it by $\hat Z.$ Finally $$Z=ρ^{-1}\dΨ\tildeΨ=\tfrac12\d u+\hat Z=:\frac12ρ^{-1}\dρ+\frac12\sum_{i\ne j}z_i^jE_j^{\ph i}.\tag{2B.7}$$

Now the second structure equation of the group of motions (2B.3b) reads $$\d(\tildeΨ E_iΨ)=\d\tildeΨ E_iΨ+\tildeΨ E_i\dΨ=ω_i^k\tildeΨ E_kΨ.\tag{2B.8}$$ Multiplying on the left by $ρ^{-1}Ψ$ and on the right by $ρ^{-1}\tildeΨ$ it becomes $$(ρ^{-1}\dΨ\tildeΨ)^\Tilde E_i+E_i(ρ^{-1}\dΨ\tildeΨ) =\tilde ZE_i+E_iZ =ω_i^kE_k.\tag{2B.9}$$ As $\hat Z$ is a bivector this is equivalent to $$ρ^{-1}\dρ E_i-[\hat Z,E_i]=ω_i^kE_k.\tag{2B.10}$$ The commutator does not cancel only for the components of $\hat Z$ in $E_i^{\ph j}$ or $E_j^{\ph i},$ which give terms in $E_j,$ so that in the whole, equating the coefficients of each $E_i$ we have $$\eqalign{ 2z_i^{j\ne i}& =ω_i^j\\ 2Z& =ρ^{-1}\dρ+\frac12\sum_{i\ne j}ω_i^jE_j^{\ph i}. }\tag{2B.11ab}$$ The second structure equation of the group of motions (2B.3b) is therefore $$\d\boldsymbol E_i=ρ^{-1}\dρ\boldsymbol E_i+2z_i^{k\ne i}\boldsymbol E_k.\tag{2B.12}$$ From here on, we no longer explicitely write conditions like $k\ne i,$ they are implied and $z^i_i$ is not defined. In pure spinor notation, this equation is equivalent to the obvious relation $$\dΨ=ZΨ,\tag{2B.GW}$$ also known as the Gauß-Weingarten equation. From the compatibility condition of the system $$\left\{\eqalign{ ∂_αΨ=Z_αΨ\\ ∂_βΨ=Z_βΨ }\right.\tag{2B.13ab}$$ as usual by cross differentiating, the scalar terms cancel and we get the so-called zero-curvature conditions $$∂_α\hat Z_β-∂_β\hat Z_α =[\hat Z_α,\hat Z_β],\tag{2B.14}$$ or in components, $$∂_α z_{i/β}^j-∂_β z_{i/α}^j =2z_{i/α}^kz_{k/β}^j-2z_{i/β}^kz_{k/α}^j.\tag{2B.15}$$ There are $\binom p2$ sets of them. In fact, expanding the second structure equation of the group of motion (2A.7b), and substituting $2z$ for $ω,$ we recover this same equation, it is the spinor expression of the integrability. This derivation is parallel to the one of the second structure equation, the cross differentiation corresponding to the exterior differentiation.

The same symmetry relations as for $ω$ can be derived directly for $Z.$ As $\hat Z$ is a bivector we have $$Z+\tilde Z=ρ^{-1}\dρ,\tag{2B.16}$$ then $$z_i^j+\eta_{ij}z_j^i=0,\quad i\ne j,\tag{2B.17}$$ so that $$z_i^jE_j^{\ph i}=z_j^iE_i^{\ph j},\tag{2B.18}$$

For separating the extrinsic and intrinsic parts, we use the involution represented by $E_P,$ and we define $$\hat Z^+=\tfrac12\{\hat Z,E_P\}E_P^{-1},\quad Z^+=\hat Z^++\tfrac12\d u,\quad Z^-=\hat Z^-=\tfrac12[Z,E_P]E_P^{-1}\tag{2B.19}$$ satisfying $$Z=Z^++Z^-,\tag{2B.20}$$ $$[\hat Z^+,E_P]=[Z^+,E_P]=\{\hat Z^-,E_P\}=0,\tag{2B.21}$$ and from the Jacobi identities $$[ [\hat Z^+,\hat Z^+],E_P]=[ [\hat Z^-,\hat Z^-],E_P]=\{[\hat Z^+,\hat Z^-],E_P\}=0.\tag{2B.22}$$ Using these commutation relations, the zero curvature condition (2B.14) splits into $$∂_α\hat Z^+_β-∂_β\hat Z^+_α =[\hat Z^+_α,\hat Z^+_β]+[\hat Z^-_α,\hat Z^-_β],\tag{2B.GR}$$ $$∂_α\hat Z^-_β-[\hat Z^+_α,\hat Z^-_β] =∂_β\hat Z^-_α-[\hat Z^+_β,\hat Z^-_α].\tag{2B.CM}$$ The first of these equations (2B.GR) contains the Gauß and the Ricci equations, while the second one (2B.CM) contains the Codazzi-Mainardi equations. They are automatically satisfied whenever $Z$ is derived from a function $Ψ(\xi),$ so that in this case the integrability condition is only concentrated in the first structure equation of the group of motions (2A.7a), which is part of the Gauß equations. This is the strength of the moving frame method.

The compound curvature and torsion tensor is defined by $$R_{αβ}+F_{αβ} :=-∂_α\hat Z^+_β+∂_β\hat Z^+_α+[\hat Z^+_α,\hat Z^+_β] =-[\hat Z^-_α,\hat Z^-_β],\tag{2B.23}$$ where $R_{αβ}$ made of all the terms in $E^{\phν}_μ$ corresponds to $θ_{αβ},$ and $F_{αβ}$ made of all the terms in $E^{\ph b}_a$ corresponds to $\tau_{αβ}.$ We have the usual expression $$R_{αβ}+F_{αβ} =[∂_α-\hat Z^+_α,∂_β-\hat Z^+_β],\tag{2B.24}$$ so that the Jacobi identity $$[∂_γ-\hat Z^+_γ,R_{αβ}+F_{αβ}] +[∂_α-\hat Z^+_α,R_{βγ}+F_{βγ}] +[∂_β-\hat Z^+_β,R_{γα}+F_{γα}]=\boldsymbol0\tag{2B.25}$$ directly gives the second Bianchi identity $$\phantom=∂_γ(R_{αβ}+F_{αβ}) +∂_α(R_{βγ}+F_{βγ}) +∂_β(R_{γα}+F_{γα})\\ =[\hat Z^+_γ,R_{αβ}+F_{αβ}] +[\hat Z^+_α,R_{βγ}+F_{βγ}] +[\hat Z^+_β,R_{γα}+F_{γα}],\tag{2B.B2}$$ which is in two independent parts $$∂_γ R_{αβ}-[\hat Z^\parallel_γ,R_{αβ}] +∂_α R_{βγ}-[\hat Z^\parallel_α,R_{βγ}] +∂_β R_{γα}-[\hat Z^\parallel_β,R_{γα}]=\boldsymbol0,\tag{2B.B2r}$$ $$∂_γ F_{αβ}-[\hat Z^\perp_γ,F_{αβ}] +∂_α F_{βγ}-[\hat Z^\perp_α,F_{βγ}] +∂_β F_{γα}-[\hat Z^\perp_β,F_{γα}]=\boldsymbol0,\tag{2B.B2f}$$ if $\hat Z^\parallel$ is made of the terms in $E^{\phν}_μ$ and $\hat Z^\perp$ of the terms in $E^{\ph b}_a.$

Finally, the fundamental forms become $${\rm I}=-\langle\d\boldsymbol X\d\boldsymbol X\rangle =ρ^2\sum_μ\eta_μω^μω^μ,\tag{2B.I}$$ $${\rm II}_a=-\langle\d(ρ^{-1}\boldsymbol E_a)\d\boldsymbol X\rangle =2ρ\sum_μ\eta_μ z^μ_{a/α}\d\xi^αω^μ,\tag{2B.II}$$ $${\rm III}_{ab}=-\langle\d(ρ^{-1}\boldsymbol E_a)\d(ρ^{-1}\boldsymbol E_b)\rangle =4\sum_μ z^μ_{a/α}z^μ_{b/β}\d\xi^α\d\xi^β.\tag{2B.III}$$

§3. Dirac equation

Now we investigate the relation between an immersed submanifold into an Euclidean space and a Dirac-like equation, using the moving frame in the language of Clifford algebras.

3A. Conformally flat submanifold

The purpose is to extend the known results for a surface [I] to any immersed $p$-manifold with $p>2.$ The former case is much simpler because every regular surface admits a conformal coordinates system, so that the $1$-forms $ω^μ$ are constant multiples of $\d\xi^μ.$ This is not true in higher dimensions. We then begin with an immersed conformally flat, or conformal for short, manifold.

Given a $p$-dimensional conformal manifold immersed in $\Bbb R^n$ with a conformal coordinates system $(\xi^1,\cdots,\xi^p),$ we have a moving frame $\boldsymbol e_i,$ defined up to a rotation in the normal space. The function $Ψ: {\cal D}\toΓ_n^0$ is then given. In this section, indices like $α,$ $β,$ $γ,…$ are all distinct.

The differential forms are chosen as $$\left[\eqalign{ ω^μ& =ρ'~\d\xi^μ,\\ ω^a& =0, }\right.\tag{3A.1}$$ where $ρ'$ is first taken arbitrary. The remaining forms are obtained from the first structure equations of the group of motions (2A.7a) that expands to $$\left[\eqalign{ ρ'^{-1}∂_νρ'~\d\xi^ν∧\d\xi^μ & =ω^μ_{ν/α}~\d\xi^ν∧\d\xi^α,\\ 0& =ω^a_{ν/α}~\d\xi^ν∧\d\xi^α, }\right.\tag{3A.2}$$ in addition to $$ω_{μ/α}^μ=ρ^{-1}∂_αρ.\tag{3A.3}$$ from (2A.6). Comparing the coefficients of the independent basis $2$-forms $\d\xi^α∧\d\xi^β$ immediately gives $$ω^γ_{α/β}=ω^γ_{β/α},\tag{3A.4}$$ $$ω^β_{α/β}=(ρρ')^{-1}∂_α(ρρ'),\tag{3A.5}$$ $$ω_{α/β}^a=ω_{β/α}^a.\tag{3A.6}$$ Then from (3A.4) we further deduce $$ω^γ_{α/β}=0\tag{3A.7}$$ by making use of the relation (2A.6) $ω^β_α=-\eta_{αβ}ω^α_β.$ Indeed this is shown by taking the cyclic permutation of the indices and subtracting two of the obtained equations from the third, or by exchanging the indices alternatively in six steps.

We proceed by calculating $$\dslashΨ:=E^μ∂_μΨ =\tfrac12E^μ\left[ρ^{-1}∂_μρ+z^j_{i/μ}E^{\ph i}_j\right]Ψ\\ =E^α\left[\tfrac12ρ^{-1}∂_αρ +z^α_{β/α}E^{\phβ}_α +z^α_{a/α}E^{\ph a}_α +\tfrac12z^β_{γ/α}E^{\phγ}_β +z^β_{a/α}E^{\ph a}_β +\tfrac12z^b_{a/α}E^{\ph a}_b\right]Ψ.\tag{3A.8}$$ The term in $z^β_{a/α}$ vanishes by symmetry, and we insert the expressions (3A.5, 3A.7) $$\dslashΨ=\left\{E^α\left[\tfrac12ρ^{-1}∂_αρ +\tfrac12(ρρ')^{-1}∂_β(ρρ')E^{\phβ}_α +\tfrac12z^b_{a/α}E^{\ph a}_b\right] -{\textstyle\sum\limits_α}z^α_{a/α}E^a\right\}Ψ.\tag{3A.9}$$ The contraction of the second term yields a term similar to the first one, so that counting carefully we get $$\hspace-21pt\dslashΨ=\left\{E^μ\left[\tfrac12(2-p)ρ^{-1}∂_μρ +\tfrac12(1-p)ρ'^{-1}∂_μρ' +\tfrac12z^b_{a/μ}E^{\ph a}_b\right] -{\textstyle\sum\limits_\alpha}z^\alpha_{a/\alpha}E^a\right\}Ψ.\tag{3A.10}$$ Now if we set $$ρ'=ρ^{(p-2)/(1-p)},\tag{3A.11}$$ $$A_α:=\tfrac12z^b_{a/α}E^{\ph a}_b,\tag{3A.12}$$ $$M:={\textstyle\sum\limits_μ}z_{a/μ}^μ E^a=:m_aE^a,\tag{3A.13}$$ we find the Dirac-like equation $$\left\{E^μ[E∂_μ-A_μ]+M\right\}Ψ=\boldsymbol0,\tag{3A.14}$$ and the $1$-forms $$ω^μ=ρ^{(p-2)/(1-p)}\d\xi^μ.\tag{3A.15}$$ The submanifold so represented is then parameterised as $$\boldsymbol X(\xi)=\boldsymbol X(\xi_0)+ \int_γρ^{(p-2)/(1-p)}(\tildeΨ E_μΨ)\d\xi'^μ,\tag{3A.16}$$ where $γ$ stands for an arbitrary path form $\xi_0$ to $\xi.$

3B. Conserved currents

The standard procedure is used to get the conserved current. We multiply the Dirac equation (3A.15) on the left by $E_P$ then by $\tildeΨ,$ and in addition we take the reverse of the equation thus obtained : $$\tildeΨ E_P\{E^μ[∂_μ-\tfrac12z^b_{a/μ}E^{\ph a}_b]+m_aE^a\}Ψ=\boldsymbol0,\tag{3B.1a}$$ $$\tildeΨ\{[\lvec∂_μ+\tfrac12z^b_{a/μ}E^{\ph a}_b]E^μ+m_aE^a\}E_PΨ =\boldsymbol0.\tag{3B.1b}$$ The tilde over $E_P$ is dropped because it is a mere change of sign. As $E^{\ph b}_a$ commutes with both $E^μ$ and $E_P,$ either adding or subtracting these equations according to the parity of $p$ gives $$\tildeΨ E_PE^μ(∂_μΨ)+(∂_μ\tildeΨ)E_PE^μΨ=\boldsymbol0,\tag{3B.2}$$ $$∂_μ(\tildeΨ E_PE^μΨ)=\boldsymbol0.\tag{3B.2’}$$ Defining the tangent pseudo-vector currents $$\boldsymbol J^μ:=\tildeΨ E_PE^μΨ,\tag{3B.3}$$ we have the continuity equations $$∂_μ\boldsymbol J^μ=\boldsymbol0.\tag{3B.4}$$ They are the same as the Noether currents associated to the right multiplication of $Ψ$ by an element of the Clifford group, but we don’t bother to write a Lagrangian. From this conservation law, as usual taking different types of terms apart, we draw the equations $${\textstyle\sum\limits_α}z^α_{β/α}=\tfrac12ρ^{-1}∂_βρ,\tag{3B.5a}$$ $$z^a_{α/β}=z^a_{β/α},\tag{3B.5b}$$ that are equivalent to the continuity equations (3B.4). But unlike the case of a surface, if $p>2$ these currents are not vectors and cannot be used to construct a manifold.

3C. Converse

We have shown that every conformal submanifold is associated to a Dirac equation of which one solution represents it. It was the easy part, now we have to prove the converse, that all solutions $Ψ\inΓ^0$ of the Dirac equation with given functions $M$ and $A_μ$ represents a submanifold, which one may not be conformal. So we have $$\left\{E^μ[E∂_μ-\tfrac12a^b_{a/μ}E^{\ph a}_b]+m_aE^a\right\}Ψ=\boldsymbol0,\tag{3C.1}$$ from which, in addition to the ones above (3B.5), we deduce the following equations $$\eta_α z^α_{β/γ} +\eta_β z^β_{γ/α} +\eta_γ z^γ_{α/β}=0,\tag{3C.2a}$$ $$z^b_{a/μ}=a^b_{a/μ},\tag{3C.2b}$$ $${\textstyle\sum\limits_μ}z^μ_{a/μ}=m_a,\tag{3C.2c}$$

For a surface, integrability is linked to the integral along a closed loop. Then theorems give the value of this integral as a function of an integral on the surface surrounded by the loop. If it is zero, the integral along a path gives a well defined immersion function. Loosely speaking, for higher dimensions this does not work because the surface integral depends on the partial derivatives along directions normal to it. To have it work, these ones should be taken into account, but the dimensions of the two integrals don’t match. For this reason, an expansion in powers series of the coordinates have to be used so that there can be a term by term cancellation. There are different ways to achieve this, but they are deducible from one another. Investigating first the simpler case of a hypersurface ($q=1,$) we proceed by looking for $1$-forms $ω^μ$ with $ω^n=0.$ To have integrability, it suffices that they satisfy the structure equations of the space $$\left[\eqalign{ \dω^μ& =ω^ν∧ω^μ_ν\\ 0& =ω^ν∧ω^n_ν }\right.\tag{3C.3ab}$$ We introduce the components of the differential forms : $$ω^μ=:ω^μ_{/α}\d\xi^α,\quad ω^i_j=:ω^i_{j/α}\d\xi^α =2z^i_{j/α}\d\xi^α+δ^i_jρ^{-1}\dρ,\tag{3C.4}$$ with which the structure equations read $$\left[\eqalign{ ∂_αω^μ_{/β}-∂_βω^μ_{/α} & =2z^μ_{ν/β}ω^ν_{/α}+ρ^{-1}∂_βρ~ω^μ_{/α} -2z^μ_{ν/α}ω^ν_{/β}-ρ^{-1}∂_αρ~ω^μ_{/β},\\ 0& =2z^n_{ν/β}ω^ν_{/α}-2z^n_{ν/α}ω^ν_{/β}, }\right.\tag{3C.5ab}$$ and we expand the searched forms in powers of the coordinates : $$ω^μ_{/α} =ρ^{-1}(ϖ^μ_{/α}+ϖ^μ_{σ/α}\xi^σ +\tfrac12ϖ^μ_{\{σ\tau\}/α}\xi^σ\xi^\tau +\tfrac1{3!}ϖ^μ_{\{σ\tauρ\}/α} \xi^σ\xi^\tau\xi^ρ+\cdots).\tag{3C.6}$$ The prefactor is chosen to get rid of meaningless derivative terms. Mind it is not a Taylor expansion, for the coefficients are not constant. To the zeroth order we get $$\left[\eqalign{ ∂_αϖ^μ_{/β} -∂_βϖ^μ_{/α} +ϖ^μ_{α/β}-ϖ^μ_{β/α} & =2z^μ_{ν/β}ϖ^ν_{/α} -2z^μ_{ν/α}ϖ^ν_{/β},\\ 0& =2z^n_{ν/β}ϖ^ν_{/α} -2z^n_{ν/α}ϖ^ν_{/β}. }\right.\tag{3C.7ab}$$ The second equation is satisfied if $$ϖ^μ_{/α}=δ^μ_{α\phantom/}\tag{3C.8}$$ owing to (3B.5b), and this is the only hypothesis we need. The first equation then gives $$ϖ^μ_{α/β}-ϖ^μ_{β/α} =2z^μ_{α/β}-2z^μ_{β/α}.\tag{3C.9}$$ A most general solution is $$ϖ^μ_{α/β} =2z^μ_{α/β}+2s^μ_{\{αβ\}}\tag{3C.10a}$$ with $s^μ_{\{αβ\}}$ symmetric in $α,β.$ And for $ϖ^μ_{α/α},$ in all generality we can similarly take $$ϖ^μ_{α/α} =2z^μ_{α/α}+2s^μ_{\{αα\}}.\tag{3C.10b}$$ We go on to first order : $$\left[\eqalign{ ∂_αϖ^μ_{σ/β} -∂_βϖ^μ_{σ/α} +ϖ^μ_{\{ασ\}/β}-ϖ^μ_{\{βσ\}/α} & =2z^μ_{ν/β}ϖ^ν_{σ/α} -2z^μ_{ν/α}ϖ^ν_{σ/β},\\ 0& =2z^n_{ν/β}ϖ^ν_{σ/α} -2z^n_{ν/α}ϖ^ν_{σ/β}. }\right.\tag{3C.11ab}$$ The second equation is a constraint for $s^μ_{\{ν\lambda\}}$ which until now was free. It is a linear system of $p^2(p-1)/2$ equations of $p^2(p+1)/2$ unknowns. However the equations are not all independent, as is easily seen by taking $σ\neα, σ\neβ,$ adding all the equations obtained by circular permutation of $α,β,σ,$ then replacing the $ϖ$ from (3C.10a, 3C.10b). The $s$ cancel out by their symmetry and the $z$ by the flatness condition (2B.15). But they are not incompatible, the number of equations is then reduced to $p(p+1)(p-1)/3.$ The unknowns that can be chosen as parameters are of the type $s^μ_{\{σμ\}}$ and $s^γ_{\{αβ\}}+s^α_{\{βγ\}}+s^β_{\{γα\}}.$ The general solution is $$ϖ^μ_{\{ασ\}/β} =∂_βϖ^μ_{σ/α} +2z^μ_{ν/β}ϖ^ν_{σ/α} +2s^μ_{\{αβ\}σ},\tag{3C.12a}$$ and $$ϖ^μ_{\{ασ\}/α} =∂_αϖ^μ_{σ/α} +2z^μ_{ν/α}ϖ^ν_{σ/α} +2s^μ_{\{αα\}σ},\tag{3C.12b}$$ but from the symmetry of $ϖ^μ_{\{σ\tau\}/α},$ we have the conditions on $s$ $$∂_αϖ^μ_{\tau/σ} +2z^μ_{ν/α}ϖ^ν_{\tau/σ} +2s^μ_{\{σα\}\tau} =∂_αϖ^μ_{σ/\tau} +2z^μ_{ν/α}ϖ^ν_{σ/\tau} +2s^μ_{\{\tauα\}σ}.\tag{3C.13}$$ At the second order $$\hspace-44pt\left[\eqalign{ ∂_αϖ^μ_{\{σ\tau\}/β} -∂_βϖ^μ_{\{σ\tau\}/α} +ϖ^μ_{\{ασ\tau\}/β} -ϖ^μ_{\{βσ\tau\}/α} & =2z^μ_{ν/β}ϖ^ν_{\{σ\tau\}/α} -2z^μ_{ν/α}ϖ^ν_{\{σ\tau\}/β},\\ 0& =2z^n_{ν/β}ϖ^ν_{\{σ\tau\}/α} -2z^n_{ν/α}ϖ^ν_{\{σ\tau\}/β}, }\right.\tag{3C.14ab}$$ with the general solutions of the first equation (3C.14a) $$ϖ^μ_{\{ασ\tau\}/β} =∂_βϖ^μ_{\{σ\tau\}/α} +2z^μ_{ν/β}ϖ^ν_{\{σ\tau\}/α} +2s^μ_{\{α/β\}σ\tau},\tag{3C.15a}$$ and $$ϖ^μ_{\{ασ\tau\}/α} =∂_αϖ^μ_{\{σ\tau\}/α} +2z^μ_{ν/α}ϖ^ν_{\{σ\tau\}/α} +2s^μ_{\{α/α\}σ\tau}.\tag{3C.15b}$$ It is obvious that each symmetry condition (3C.13) eliminates a different equation (3C.14a), so that there is no net effect on the number of constraints. The coefficient matrix of this system is the same as above, and there are as many such systems, and as many series of unknowns as $\tau$ values. All considered, there are then $p^2(p+1)(p-1)/3$ equations and $p^3(p+1)/2$ unknowns.

We can continue to higher and higher orders, we always get the same types of terms. By choosing appropriate functions for the parameters, solutions can be found that are defined everywhere, even at the points where the matrix of coefficients is singular, and for each order. The known case of a surface is of this type, by taking $s^1_{\{21\}}=0,$ $s^2_{\{12\}}=0,$ and so on. The problem is not the lack of solution, but the plethora of them, even for a surface. But already for three dimensions the calculations become excruciating, and when $q\gt1,$ the number of equation is multiplied by $q,$ so that it does not work. To get something simpler and universal, we must relax the condition that the tangent vectors to the manifold should belong to the space spanned by the $\boldsymbol E_μ,$ or as we shall say, be parallel, since the constraint is a consequence of $ω^a=0,$ and so this is no longer true.

For that we use another, equivalent method. If we find independent vectors $\boldsymbol I_α$ such that $$∂_β\boldsymbol I_α -∂_α\boldsymbol I_β=\boldsymbol0,\tag{3C.16}$$ then there exist an immersion function $\boldsymbol X$ such that $\boldsymbol I_α=∂_α\boldsymbol X.$ On the other hand we should have $$\d\boldsymbol X=ω^i_{/α}\boldsymbol E_i\d\xi^α,\tag{3C.17}$$ so that $$\boldsymbol I_α=ω^i_{/α}\boldsymbol E_i.\tag{3C.18}$$ Next we also expand this vectors in power series $$\boldsymbol I_α=\boldsymbol K_α +\boldsymbol K_{α|σ}\xi^σ +\tfrac12\boldsymbol K_{α|\{σ\tau\}}\xi^σ\xi^\tau +\tfrac1{3!}\boldsymbol K_{α|\{σ\tauρ\}}\xi^σ\xi^\tau\xi^ρ +\cdots,\tag{3C.19}$$ with the correspondance $$\boldsymbol K_α=ϖ^i_{/α}\hat{\boldsymbol E}_i,\quad \boldsymbol K_{α|σ} =ϖ^i_{σ/α}\hat{\boldsymbol E}_i\xi^σ,\\ \boldsymbol K_{α|\{σ\tau\}} =ϖ^i_{\{σ\tau\}/α}\hat{\boldsymbol E}_i\xi^σ\xi^\tau,\quad \boldsymbol K_{α|\{σ\tauρ\}} =ϖ^i_{\{σ\tauρ\}/α}\hat{\boldsymbol E}_i\xi^σ\xi^\tau\xi^ρ,\cdots\tag{3C.20}$$ and the definition $$\hat{\boldsymbol E}_i:=ρ^{-1}\boldsymbol E_i.\tag{3C.21}$$ So we have to look for the $\boldsymbol K$ vectors. According to the foregoing, we make the first guess $$\boldsymbol K_α=\hat{\boldsymbol E}_α\tag{3C.22}$$ and evaluate (3C.16) order by order. At the zeroth one we get $$∂_β\boldsymbol K_α +\boldsymbol K_{α|β} =∂_α\boldsymbol K_β +\boldsymbol K_{β|α},\tag{3C.23}$$ $$2z^μ_{α/β}\hat{\boldsymbol E}_μ +\boldsymbol K_{α|β} =2z^μ_{β/α}\hat{\boldsymbol E}_μ +\boldsymbol K_{β|α},\tag{3C.23’}$$ and chose the solutions $$\boldsymbol K_{α|β} =2z^μ_{β/α}\hat{\boldsymbol E}_μ,\tag{3C.24a}$$ $$\boldsymbol K_{α|α} =2z^μ_{α/α}\hat{\boldsymbol E}_μ.\tag{3C.24b}$$ Again at the first order $$∂_β\boldsymbol K_{α|σ} +\boldsymbol K_{α|\{βσ\}} =∂_α\boldsymbol K_{β|σ} +\boldsymbol K_{β|\{ασ\}},\tag{3C.25}$$ $$∂_β(2z^μ_{σ/α}\hat{\boldsymbol E}_μ) +\boldsymbol K_{α|\{βσ\}} =∂_α(2z^μ_{σ/β}\hat{\boldsymbol E}_μ) +\boldsymbol K_{β|\{ασ\}},\tag{3C.25'}$$ $$∂_β(∂_α\hat{\boldsymbol E}_σ -2z^a_{σ/α}\hat{\boldsymbol E}_a) +\boldsymbol K_{α|\{βσ\}} =∂_α(∂_β\hat{\boldsymbol E}_σ -2z^a_{σ/β}\hat{\boldsymbol E}_a) +\boldsymbol K_{β|\{ασ\}},\tag{3C.25''}$$ $$-∂_β(2z^a_{σ/α}\hat{\boldsymbol E}_a) +\boldsymbol K_{α|\{βσ\}} =-∂_α(2z^a_{σ/β}\hat{\boldsymbol E}_a) +\boldsymbol K_{β|\{ασ\}},\tag{3C.25'''}$$ and we can take the solutions $$\boldsymbol K_{α|\{σ\tau\}} =-∂_α(2z^a_{σ/\tau}\hat{\boldsymbol E}_a),\tag{3C.26}$$ which have the correct symmetry. At the second order something happens : $$∂_β\boldsymbol K_{α|\{σ\tau\}} +\boldsymbol K_{α|\{βσ\tau\}} =∂_α\boldsymbol K_{β|\{σ\tau\}} +\boldsymbol K_{β|\{ασ\tau\}},\tag{3C.27}$$ $$-∂_{βα}(2z^a_{σ/\tau}\hat{\boldsymbol E}_a) +\boldsymbol K_{α|\{βσ\tau\}} =-∂_{αβ}(2z^a_{σ/\tau}\hat{\boldsymbol E}_a) +\boldsymbol K_{β|\{ασ\tau\}},\tag{3C.27'}$$ as due to the cancellation we have $$\boldsymbol K_{α|\{σ\tauρ\}} =\boldsymbol0,\tag{3C.28}$$ and similarly for the remaining orders. Finally we have found the finite and regular series $$\boldsymbol I_α =\hat{\boldsymbol E}_α +2z^μ_{σ/α}\hat{\boldsymbol E}_μ\xi^σ -∂_α(z^a_{σ/\tau}\hat{\boldsymbol E}_a)\xi^σ\xi^\tau,\tag{3C.29}$$ with which (3C.16) can be verified directly. We have proven the converse provided there is no restriction on the value of the tangent vectors. This solution is not the same as the one given above for a conformal manifold, even for the same solution of the Dirac equation.

To conclude this section, under the previous assumption that $Ψ$ is everywhere an element of the special Clifford group, a submanifold is represented by every solution of a Dirac equation, whatever the dimensions and the signatures. In addition for a hypersurface, under the proviso that the series converges and except for possible isolated singular points, this is also true with parallel tangent vectors. As we used the only hypothesis (3B.5b), this can be viewed as the direct consequence of the current conservation alone.

§4. Extended Gauß map

4A. Definition

Suppose the local tangent space of the immersed submanifold is spanned by $\{\boldsymbol E_μ\}.$ Then its normal vectors are the $\boldsymbol E_a.$ In the Clifford algebra, the normal space can be represented by a single object : $\boldsymbol E_{p+1}\cdots\boldsymbol E_n=\boldsymbol E_{p+1\cdots n}.$ It is equivalent and more convenient to use its dual, the pseudo-scalar $\boldsymbol E_1\cdots\boldsymbol E_p=\boldsymbol E_P$ of the immersed manifold, since this avoids the reference to the ambient space. Then in the general case, the pseudo-scalar is $$\tildeΨ E_1Ψ\cdots\tildeΨ E_pΨ=ρ^{p-1}\tildeΨ E_PΨ.\tag{4A.1}$$ Since we are only interested in the direction, the norm is immaterial, so we shall take $$\hat{\boldsymbol E}_P:=ρ^{-1}\tildeΨ E_PΨ =Ψ^{-1}E_PΨ.\tag{4A.2}$$ We call this the extended Gauß map. (We reserve the term generalised Gauß map to undefined quadratic forms in order to avoid confusions, since it is used indiscriminately in the literature.) From this expression we see that multiplying $Ψ$ on the left by an element of the group ${\rm Lor}(p)={\rm Spin}(p)$ whose Lie algebra is $\span\{E_{μν}/2\},$ or of the group ${\rm YM}(q)={\rm Spin}(q)$ whose Lie algebra is $\span\{E_{ab}/2\},$ the Gauß map is unaffected, so that it is an element of the homogeneous space of ${\rm Spin}(n)$ $${{\rm Spin}(n)\over{\rm Spin}(p)\times{\rm Spin}(q)} \sim{{\rm SO}(n)\over{\rm SO}(p)\times{\rm SO}(q)},\tag{4A.3}$$ It is the real oriented Grassmann manifold $\tilde{\rm G}_{p,q},$ while ${\rm Lor}(p)\times{\rm YM}(q)$ is the isotropy group. (Like $n,$ $p$ and $q$ are understood as $p_+,p_-$ and $q_+,q_-.$) Because of the invariance under ${\rm Lor}(p)\times{\rm YM}(q),$ the Gauß map can be represented by $$\hat{\boldsymbol E}_P=\exp\{-U/2\}E_P\exp\{U/2\}=R^{-1}E_P=E_PR\tag{4A.4}$$ with the notations $$U:=u^{μ a}E_{μ a},\quad R:=\e^U.\tag{4A.5}$$ Globally, the extended Gauß map has $n(n-1)/2$ real components, but they satisfy $[p(p-1)+q(q-1)]/2$ quadratic constraints. The real dimension of the Gauß map manifold, i.e. the range of the Gauß map, is then $$\tfrac12[n(n-1)-p(p-1)-q(q-1)]=pq.\tag{4A.6}$$ Locally, it is represented by $\Bbb W=\span\{E_{μ a}/2\},$ which is the differential of the coset space and transforms under the adjoint representation of the isotropy group. We then have the linear space decomposition $${\frak spin}(n)={\frak lor}(p)\oplus\Bbb W\oplus{\frak ym}(q),\tag{4A.7}$$ with $$[{\frak lor},{\frak lor}]\subset{\frak lor},\quad [{\frak ym},{\frak ym}]\subset{\frak ym},\quad [{\frak lor},{\frak ym}]=\{\boldsymbol0\},\\ [{\frak lor},\Bbb W]\subset\Bbb W,\quad [{\frak ym},\Bbb W]\subset\Bbb W,\quad [\Bbb W,\Bbb W]\subset{\frak lor}\oplus{\frak ym}.\tag{4A.8}$$

4B. Stereographic projection

It is always interesting to stereographically project the Gauß map onto $\Bbb W.$ The definition of this extended stereographic projection requires a foray into the realm of orthogonal matrices. It is essentially a function from an orthogonal group to its Lie algebra, and is actually a restriction of the inverse Cayley transform.

Because a ${\rm SO}(n)$ matrix is block-diagonalisable, it can be split into the product of mutually commuting matrices. So let ${\cal R}$ be the matrix of ${\rm SO}(n),$ and let ${\cal M}$ be the orthogonal matrix that diagonalises it : $${\cal MRM}^{-1}= {\small\pmatrix{\cosθ_1& -\sinθ_1& 0& 0& \cdots\\ \sinθ_1& \phantom-\cosθ_1& 0& 0& \cdots\\ 0& 0& \cosθ_2& -\sinθ_2& \dots\\ 0& 0& \sinθ_2& \phantom-\cosθ_2& \dots\\ \vdots& \vdots& \vdots& \vdots& \ddots}}\\ ={\small\pmatrix{\cosθ_1& -\sinθ_1& 0& 0& \cdots\\ \sinθ_1& \phantom-\cosθ_1& 0& 0& \cdots\\ 0& 0& 1& 0& \dots\\0& 0& 0& 1& \dots\\ \vdots& \vdots& \vdots& \vdots& \ddots} \pmatrix{1& 0& 0& 0& \cdots\\0& 1& 0& 0& \cdots\\ 0& 0& \cosθ_2& -\sinθ_2& \dots\\ 0& 0& \sinθ_2& \phantom-\cosθ_2& \dots\\ \vdots& \vdots& \vdots& \vdots& \ddots}}\cdots =:{\cal R}'_1{\cal R}'_2\cdots\tag{4B.1}$$ Then $${\cal R=M}^{-1}{\cal R}'_1{\cal R}'_2\cdots {\cal M =(M}^{-1}{\cal R}'_1{\cal M)(M}^{-1}{\cal R}'_2{\cal M})\cdots =:{\cal R}_1{\cal R}_2\cdots\tag{4B.2}$$ For non compact groups ${\rm SO}(n_+,n_-),$ the blocks may look like $$\pmatrix{\cosh(θ)& \sinh(θ)\\\sinh(θ)& \cosh(θ)},\tag{4B.3}$$ but mutatis mutandis, the reasoning and the results are the same.

The inverse Cayley transform is a projection of ${\cal R}$ from an orthogonal group ${\rm SO}(n)$ to ${\cal W}$ in its Lie algebra ${\frak so}(n)$ given by one of these equal expressions : $${\cal W}:={{\cal R}-{\cal E}\over{\cal R}+{\cal E}} ={{\cal R}-{\cal R}^{-1}\over2{\cal E}+{\cal R}+{\cal R}^{-1}} ={{\cal R}^{1/2}-{\cal R}^{-1/2}\over{\cal R}^{1/2}+{\cal R}^{-1/2}},\tag{4B.4abc}$$ where ${\cal E}$ is the unit matrix. The numerator and the denominator commute, which justifies the fraction notation. We take the last expression because cancellations will occur. Suppose first that ${\cal R}$ is block-diagonal. Then we have $$\begin{multline}\hspace{3em} {\cal W}={\small\pmatrix{0& -\sin(θ_1/2)& 0& 0& \cdots\\ \sin(θ_1/2)& 0& 0& 0& \cdots\\ 0& 0& 0& -\sin(θ_2/2)& \dots\\ 0& 0& \sin(θ_2/2)& 0& \dots\\ \vdots& \vdots& \vdots& \vdots& \ddots}}\times\\ {\small\pmatrix{\cos(θ_1/2)& 0& 0& 0& \cdots\\ 0& \phantom-\cos(θ_1/2)& 0& 0& \cdots\\ 0& 0& \cos(θ_2/2)& 0& \dots\\ 0& 0& 0& \phantom-\cos(θ_2/2)& \dots\\ \vdots& \vdots& \vdots& \vdots& \ddots}}^{\Large -1}\\ =\small\pmatrix{0& -\tan(θ_1/2)& 0& 0& \cdots\\ \tan(θ_1/2)& 0& 0& 0& \cdots\\ 0& 0& 0& -\tan(θ_2/2)& \dots\\ 0& 0& \tan(θ_2/2)& 0& \dots\\ \vdots& \vdots& \vdots& \vdots& \ddots}. \hspace{3em}\tag{4B.5}\end{multline}$$ It is clear that ${\cal W}$ is block-diagonal too, and is a sum $${\cal W}=\sum_i{\cal W}_i\tag{4B.6}$$ in which the ${\cal W}_i$ contain only the $i$-th block and $0$ everywhere else. Now we can write them as the inverse Cayley transform of ${\rm SO}(n)$ matrices ${\cal R}_i$ containing the corresponding block, $1$ on the remaining diagonal and $0$ everywhere else. It would be cumbersome to write it explicitly, so it is left as an exercice. The key is that in the numerator the diagonal $1$’s cancel out. As seen above, the product of all these matrices is just ${\cal R}$ : $${\cal R}=\prod_i{\cal R}_i.\tag{4B.7}$$ So we arrive at $${\cal W} =\sum_i{{\cal R}_i^{1/2}-{\cal R}_i^{-1/2}\over{\cal R}_i^{1/2}+{\cal R}_i^{-1/2}} ={\sum_i({\cal R}_i^{1/2}-{\cal R}_i^{-1/2}) \prod_{j\ne i}({\cal R}_j^{1/2}+{\cal R}_j^{-1/2}) \over\prod_k({\cal R}_k^{1/2}+{\cal R}_k^{-1/2})}.\tag{4B.8}$$ If ${\cal R}$ is not block-diagonal, it can be written as $${\cal R}={\cal M}^{-1}{\cal R}'{\cal M}=\prod_i{\cal M}^{-1}{\cal R'}_i{\cal M},\tag{4B.9}$$ where ${\cal R}'$ is so, and ${\cal M}$ is an orthogonal matrix, then $${\cal M}^{-1}{\cal W'}{\cal M} ={\cal M}^{-1}\sum_i{{\cal R'}_i^{1/2}-{\cal R'}_i^{-1/2} \over{\cal R'}_i^{1/2}+{\cal R'}_i^{-1/2}}{\cal M},\tag{4B.10}$$ which is the same formula as above (4B.8). Expanding the products, remarking with the matrices that $${\cal R}_i^{1/2}{\cal R}_j^{-1/2}+{\cal R}_i^{-1/2}{\cal R}_j^{1/2} ={\cal R}_i^{1/2}{\cal R}_j^{1/2}+{\cal R}_i^{-1/2}{\cal R}_j^{-1/2},\tag{4B.11}$$ the initial expression (4B.4c) is recovered. This additive property of the Cayley transform seems to be unknown.

Back in the Clifford algebras, the bivector $U$ similarly decomposes into a sum of orthogonal simple bivectors $$U=\sum_iU_i,\quad U_iU_i=\pm(θ_i)^2E,\quad[U_i,U_j]=\boldsymbol0,\tag{4B.12}$$ each corresponding to the $W_i,$ so that we get $$W=\sum_i W_i,\quad [W_i,W_j]=\boldsymbol0,\tag{4B.13}$$ $$W=\sum_i{\e^{U_i/2}-\e^{-U_i/2}\over\e^{U_i/2}+\e^{-U_i/2}} ={\sum_i[(\e^{U_i/2}-\e^{-U_i/2})\prod_{j\ne i}(\e^{U_j/2}+\e^{-U_j/2})] \over\prod_k(\e^{U_k/2}+\e^{-U_k/2})}.\tag{4B.14}$$ The expansion of the numerator can be written as $$\begin{multline}\hspace{6em} n(\e^{U/2}-\e^{-U/2})+(n-2)\sum_{i=1}^n(\e^{U/2-U_i}-\e^{-U/2+U_i})+\\ +(n-4)\sum_{i\lt j}^n(\e^{U/2-U_i-U_j}-\e^{-U/2+U_i+U_j})+\cdots \hspace{6em}\tag{4B.15}\end{multline}$$ for the coefficients $(n-2k)\gt0,$ which arises from cancellation of identical terms because of the minus sign. And the denominator is $$\hspace-39pt(\e^{U/2}+\e^{-U/2})+\sum_{i=1}^n(\e^{U/2-U_i}+\e^{-U/2+U_i}) +\sum_{i\lt j}^n(\e^{U/2-U_i-U_j}+\e^{-U/2+U_i+U_j})+\cdots,\tag{4B.16}$$ with $[n/2]+1$ terms. Finally, by expanding the exponential functions all the products $U_iU_j\cdots$ cancel out, leaving only the bivector or scalar terms, so we see that this is equal to the compact form $$W={\langle\e^{U/2}-\e^{-U/2}\rangle_2\over\langle\e^{U/2}+\e^{-U/2}\rangle} ={\langle\e^{U/2}\rangle_2\over\langle\e^{U/2}\rangle},\tag{4B.17}$$ where the decomposition also does not appear explicitly.

The parameters of $W$ consist of $\min(p,q)$ polar angles, or moduli when they are projected, and the others are azimutal angles contained in the diagonalizing orthogonal rotation. The closed $\Bbb W$ plane has then at most $\min(p,q)$ distinct infinities $\infty_i,$ corresponding to $θ_i=\pi.$

The direct Cayley transform is given by $${\cal R}={{\cal E+W\over E-W}}.\tag{4B.18}$$ This expression gives the Clifford equivalent Lipschitz chart $$\e^U=\prod_i\e^{U_i}=\prod_i{E+W_i\over E-W_i}={\e^{∧ W}\over\e^{∧ -W}},\tag{4B.19}$$ $$\e^{U/2}={\e^{∧ W}\over|\e^{∧ W}|},\tag{4B.20}$$ where $$\eqalign{ \e^{∧ W}:& =E+W+\tfrac12W∧ W+\tfrac1{3!}W∧ W∧ W+\cdots\\ :& =\langle E\rangle_0+\langle W\rangle_2+\tfrac12\langle WW\rangle_4+\tfrac1{3!}\langle WWW\rangle_6+\cdots\\ & =E+\sum_i W_i+\sum_{i\lt j}W_iW_j+\sum_{i\lt j\lt k}W_iW_jW_k+\cdots, }\tag{4B.21}$$ $$|\e^{∧ W}|^2=\e^{∧ W}\e^{∧ -W}={\textstyle\prod_i}(E-W_iW_i).\tag{4B.22}$$ This result is given in [L’01 §17.3]. Putting this expression into the inverse transform (4B.17) gives back $W,$ which provides an independent check of the latter.

4C. Lift

In the appropriate gauge, we can locally write $Ψ$ simply as follow : $$Ψ=:\pmρ^{1/2}\e^{V/2}\e^{U/2},\quad V=:v^{αβ}E_{αβ}.\tag{4C.1}$$ Then the Gauß map becomes $$\hat{\boldsymbol E}_P=Ψ^{-1}E_PΨ=E_P\e^U,\tag{4C.2}$$ so that we have the stereographic projection and back $$W={\langle(E_P^{-1}\hat{\boldsymbol E}_P)^{1/2}\rangle_2 \over\langle(E_P^{-1}\hat{\boldsymbol E}_P)^{1/2}\rangle},\quad \hat{\boldsymbol E}_P=E_P{\e^{∧ W}\over\e^{∧-W}}.\tag{4C.3ab}$$ Now let us consider only normal spaces that are Euclidean. Although $V$ obviously cannot be expressed algebraically from $W$ like $U,$ it can differentially be so through the Dirac equation supposing that $Ψ$ is a solution. We write it as $$\hspace-19pt\dslashΨΨ^{-1}=\tfrac12ρ^{-1}\dslashρ+(\dslash\e^{V/2})\e^{-V/2} +E^μ\e^{V/2}(∂_μ\e^{U/2})\e^{-U/2}\e^{-V/2}=-M+E^μ A_μ.\tag{4C.4}$$ We are only interested in the terms with $E^a.$ There is none in the first and second term, then we get $$E^μ\e^{V/2}(∂_μ\e^{U/2})\e^{-U/2}\e^{-V/2}=-m_aE^a+E^μ a^b_{a/μ}E^{\ph a}_b,\tag{4C.5}$$ where $m_a$ and $a^a_{b/μ}$ are given functions. Considering the expansion [M72, Lemma 5.4] $$(\d\e^X)\e^{-X}=\d X+\tfrac12[X,\d X]+\tfrac1{3!}[X,[X,\d X]]+\cdots\tag{4C.6}$$ with the commutation rules (4A.8) and the fact that $U\in\Bbb W,$ all the remaining terms with no $E^a$ as well as with $E^{\ph a}_b$ are eliminated by writing $$E^μ\e^{V/2}[(∂_μ\e^{U/2})\e^{-U/2}-(∂_μ\e^{-U/2})\e^{U/2}]\e^{-V/2}=-2m_aE^a\tag{4C.7}$$ instead, where the potential term also disappears. From the expression (4C.3b) we calculate $$(∂_μ\e^{U/2})\e^{-U/2} ={(∂_μ\e^{∧ W})\e^{∧ -W} -|\e^{∧ W}|∂_μ|\e^{∧ W}| \over|\e^{∧ W}|^2},\tag{4C.8}$$ leading to the new equation $$E^μ\e^{V/2}[(∂_μ\e^{∧ W})\e^{∧ -W} -(∂_μ\e^{∧ -W})\e^{∧ W}]\e^{-V/2} =-2|\e^{∧ W}|^2m_aE^a.\tag{4C.9}$$ In the case of a curve ($p=1$) or of a hypersurface ($q=1,$) this is greatly simplified since $W$ is simple (the others are reputed to be algebraically arduous,) the equation then becomes $$E^μ\e^{V/2}(∂_μ W)\e^{-V/2} =-(E-WW)m_aE^a.\tag{4C.10}$$ It looks like a trivial equation, but in general there is no simple commutation relation between any of the factors. Already for $p\geqslant3$ it is intractable. However again, for a surface and any codimension, it is simpler since $V$ anti-commutes with $E^μ,$ $W,$ and its derivatives. So we divide (4C.9) by its reverse $$\e^{V/2}[\e^{∧ W}(∂_μ\e^{∧ -W}) -\e^{∧ -W}(∂_μ\e^{∧ W})]\e^{-V/2}E^μ =-2|\e^{∧ W}|^2m_aE^a,\tag{4C.11}$$ to get $$\begin{multline}\hspace{6em} E^μ\e^{V/2}[(∂_μ\e^{∧ W})\e^{∧ -W} -(∂_μ\e^{∧ -W})\e^{∧ W}]\e^{-V/2}\times\\ \{[\e^{∧ W}(∂_ν\e^{∧ -W}) -\e^{∧ -W}(∂_ν\e^{∧ W})]\e^{-V/2}E^ν\}^{-1}\e^{-V/2}=E, \hspace{6em}\tag{4C.12}\end{multline}$$ which through the anti-commutations gives $$\e^{2V} =[(\dslash\e^{∧ W})\e^{∧ -W} -(\dslash\e^{∧ -W})\e^{∧ W}] [\e^{∧ W}(\e^{∧ -W}\lvec\dslash) -\e^{∧ -W}(\e^{∧ W}\lvec\dslash)]^{-1}.\tag{4C.13}$$

We obtain a further relation by multiplying the two equations : $$\begin{multline}\hspace{6em} E^μ\e^{V/2}[(∂_μ\e^{∧ W})\e^{∧ -W} -(∂_μ\e^{∧ -W})\e^{∧ W}]\times\\ [\e^{∧ W}(∂_ν\e^{∧ -W}) -\e^{∧ -W}(∂_ν\e^{∧ W})]\e^{-V/2}E^ν =4|\e^{∧ W}|^4m^2, \hspace{6em}\tag{4C.14}\end{multline}$$ $$m^2=m^am_a,\tag{4C.15}$$ that is, for $W$ simple $$E^μ\e^{V/2}(∂_μ W)(W\lvec∂_ν)\e^{-V/2}E^ν =(E-WW)^2m^2,\tag{4C.16}$$ and for a surface $$\hspace-9pt[(\dslash\e^{∧ W})\e^{∧ -W} -(\dslash\e^{∧ -W})\e^{∧ W}] [\e^{∧ W}(\e^{∧ -W}\lvec\dslash) -\e^{∧ -W}(\e^{∧ W}\lvec\dslash)] =4|\e^{∧ W}|^4m^2.\tag{4C.17}$$

Having $U$ and $V,$ we can look for $ρ,$ following the same strategy. From the Dirac equation we get $$\hspace-18pt\tfrac12ρ^{-1}\dslashρ+(\dslash\e^{V/2})\e^{-V/2} +E^μ\e^{V/2}[(∂_μ\e^{U/2})\e^{-U/2} +(∂_μ\e^{-U/2})\e^{U/2}]_\parallel\e^{-V/2}=\boldsymbol0,\tag{4C.18}$$ where the symbol $\parallel$ means that we keep only the tangent terms, that is, in $E_{μν}.$ Under the condition that this system be compatible, $ρ$ is determined by integration up to an overall multiplicative constant. It is a necessary condition for $Ψ$ satisfying the Dirac equation to exist. If it does, it represents a submanifold whose Gauß map is $\hat{\boldsymbol E}_P.$ Given a submanifold with its coordinate system, the Gauß map is known, a function $Ψ\inΓ^0$ of these coordinates could then be found that on one hand is a solution of a Dirac equation, and on the other hand is such that the vectors $\boldsymbol E_μ=\tildeΨ E_μΨ$ are tangent to the submanifold. To any submanifold would then be associated a Dirac equation and a distinguished solution thereof.

§5. Known representations

To obtain the already known formulæ as particular cases, a matrix representation of the various Clifford algebra must be chosen. We begin with the simplest case.

5A. Curves

Two dimensions

For $p=1$ and $q=1,$ the Dirac equation is $$\{E^1∂_1+mE^2\}Ψ=\boldsymbol0.\tag{5A.1}$$ It is put in its conventional form by multiplying by $E_2$ on the left $$\{E^{\ph1}_2∂_1-mE\}Ψ=\boldsymbol0.\tag{5A.2}$$

Curve in $\Bbb R^2$

The very first case is a plane curve in $\Bbb R^2,$ then we need a representation of ${\rm C}\ell_{2,0}$ : $$E=\boldsymbol 1,\quad E_1=iσ_1,\quad E_2=iσ_2,\quad E_{12}=-iσ_3,\tag{5A.3}$$ where we use the standard Pauli matrices $$\boldsymbol 1=\pmatrix{1& 0\\0& 1},\quad σ_1=\pmatrix{0& \phantom-1\\1& \phantom-0},\quad σ_2=\pmatrix{0& -i\\i& \phantom-0},\quad σ_3=\pmatrix{1& \phantom-0\\0& -1}.\tag{5A.4}$$ An element of the special Clifford group is given by $$Ψ=ψ E-iψ^{12}σ_3 =ρ^{1/2}\exp\{+i\phi\tfrac12σ_3\},\tag{5A.5}$$ $$\tilde Ψ=ψ E+iψ^{12}σ_3 =ρ^{1/2}\exp\{-i\phi\tfrac12σ_3\}.\tag{5A.$\tilde5$}$$ viz. $$Ψ=ρ^{1/2}\pmatrix{\e^{i\phi/2}& 0\\0& \e^{-i\phi/2}},\quad \tildeΨ=ρ^{1/2}\pmatrix{\e^{-i\phi/2}& 0\\0& \e^{i\phi/2}}\tag{5A.6}$$ The Dirac equation (5A.2) is represented as $$\{iσ^3∂_1-m\boldsymbol1\}Ψ=\boldsymbol0,\tag{5A.7}$$ and reduces to $$∂_1ρ^{1/2}=0,\quad ∂_1\phi=2m.\tag{5A.8}$$ As $ρ$ is constant, with no loss of generality we can set it to $1.$ The tangent vector is $$\boldsymbol E_1=\tildeΨ E_1Ψ=E_1\exp\{i\phiσ_3\} =\pmatrix{0& i\e^{-i\phi}\\i\e^{i\phi}& 0} =\cos(\phi)E_1+\sin(\phi)E_2,\tag{5A.9}$$ then $2m$ is the curvature since it is the variation of the angle $\phi$ between $\boldsymbol E_1$ and $E_1.$ Without going into the details, it is clear that $\boldsymbol E_1$ gives a direction at each $\xi^1,$ which is the arclength parameter, and we know that in one dimension it is always integrable. There is no difficulty in deriving the induction formula.

Three dimensions

For $p=1$ and $q=2,$ the Dirac equation is $$\{E^1[E∂_1-\tfrac12aE^{\ph3}_2]+m_2E^2+m_3E^3\}Ψ=\boldsymbol0.\tag{5A.10}$$ It is always possible to perform a gauge transformation so that $m_3=0,$ and we write $m=m_2.$ Multiplying on the left by $E_2$ it becomes $$\{E^{\ph1}_2[E∂_1-\tfrac12aE^{\ph3}_2]-mE\}Ψ=\boldsymbol0.\tag{5A.11}$$

Curve in $\Bbb R^3$

For ${\rm C}\ell_{3,0}$ we take the representation $$E=\boldsymbol1⊗\boldsymbol1,\quad E_i=σ_3⊗ iσ_i,\quad E_{jk}=-\boldsymbol1⊗ iσ_i,\quad E_{123}=σ_3⊗\boldsymbol1\tag{5A.12}$$ Here and thereafter $i,j,k$ is a cyclic permutation of $1,2,3,$ and $⊗$ is the Kronecker product of matrices with the properties $$({\sf a⊗ x})({\sf b⊗ y})=({\sf ab})⊗({\sf xy}),\\ \bTr{\sf a⊗ x}=\bTr{\sf a}\bTr{\sf x},\\ {\sf a}⊗({\sf x+y})={\sf a⊗ x+a⊗ y},\quad ({\sf a+b})⊗{\sf x=a⊗ x+b⊗ x},\\ (\lambda{\sf a})⊗{\sf x=a}⊗(\lambda{\sf x})=\lambda({\sf a⊗ x}).\tag{5A.13}$$ The spinor is given by $$Ψ=ψ E+ψ^{jk}E_{jk} =\boldsymbol 1⊗(ψ-ψ^{jk}iσ_i)=:\boldsymbol 1⊗\Phi\tag{5A.14}$$ $$\tildeΨ=ψ E-ψ^{jk}E_{jk}=\boldsymbol 1⊗\Phi^†\tag{5A.$\tilde{14}$}$$ where $$\Phi=:\pmatrix{φ& -\barχ\\χ& \phantom-\barφ}\in \Bbb R\times {\rm SU}(2),\tag{5A.15}$$ and $$\Phi^†\Phi=ρ\boldsymbol 1,\quad \tildeΨΨ=ρ E.\tag{5A.16}$$ The Dirac equation (5A.11) is represented as $$\{\boldsymbol1⊗ iσ^3[\boldsymbol1⊗\boldsymbol1∂_1 +\tfrac12a\boldsymbol1⊗ iσ^1]-m\boldsymbol1⊗\boldsymbol1\}(\boldsymbol1⊗\Phi)=0.\tag{5A.16}$$ Dropping the redundant $\boldsymbol1⊗$ factor we have $$\{iσ^3[\boldsymbol1∂_1 +\tfrac12a iσ^1]-m\boldsymbol1\}\Phi=\boldsymbol0.\tag{5A.17}$$ It directly gives the Maurer-Cartan form $$Z_1=ρ^{-1}∂_1Ψ\tildeΨ=-\boldsymbol1⊗(\tfrac12a iσ_1+miσ_3).\tag{5A.18}$$ As there is no scalar term, $ρ$ is anew constant and set to $1.$ The tangent vector is $$\boldsymbol E_1 =\tildeΨ(σ_3⊗ iσ_1)Ψ =(\boldsymbol1⊗\Phi^†) (σ_3⊗ iσ_1)(\boldsymbol1⊗\Phi) =σ_3⊗\Phi^† iσ_1\Phi.\tag{5A.19}$$ With this we calculate $$∂_1\boldsymbol E_1 =\tildeΨ[Z_1,σ_3⊗ iσ_1]Ψ =2m\tildeΨ\boldsymbol(σ_3⊗ iσ_2)Ψ=2m\boldsymbol E_2.\tag{5A.20}$$ In this gauge, $\boldsymbol E_2$ is the normal vector of the Frenet frame, therefore $\boldsymbol E_3$ is the binormal vector, and the curvature is $2m.$ Similarly we get $$∂_1\boldsymbol E_3=-a\boldsymbol E_2,\tag{5A.21}$$ the torsion is $a.$ And $$∂_1\boldsymbol E_2=a\boldsymbol E_3-2m\boldsymbol E_1,\tag{5A.22}$$ which is the remaining Frenet-Serret equation.

5B. Surfaces

Three dimensions

For $p=2$ and $q=1,$ the Dirac equation is $$\{E^1∂_1+E^2∂_2+mE^3\}Ψ=\boldsymbol0,\tag{5B.1}$$ and in the conventional form $$\{E^{\ph1}_3∂_1+E^{\ph2}_3∂_2-mE\}Ψ=\boldsymbol0.\tag{5B.2}$$

Surface in $\Bbb R^3$

The representation of the Clifford algebra and the spinor are the sames (5A.12, 5A.14) as above. The Dirac equation (5B.2) is represented as $$\left\{-\boldsymbol1⊗ iσ^2∂_1+\boldsymbol1⊗ iσ^1∂_2 -m\boldsymbol1⊗\boldsymbol1\right\}(\boldsymbol1⊗\Phi)=\boldsymbol0,\tag{5B.3}$$ and dropping the redundant $\boldsymbol1⊗$ left factor we get the usual equation [I] $$\left\{-iσ^2∂_1+iσ^1∂_2-m\boldsymbol1\right\}\Phi=\boldsymbol0.\tag{5B.4}$$ With the definition of complex variables and derivatives $$ζ:=\xi^1+i\xi^2,\tag{5B.5}$$ $$∂:=∂_ζ=\tfrac12(∂_1-i∂_2),\quad \bar∂:=∂_\barζ=\tfrac12(∂_1+i∂_2),\tag{5B.6}$$ it splits into $$\left\{\eqalign{ \bar∂φ=\phantom-\tfrac12mχ,\\ ∂χ=-\tfrac12mφ. }\right.\tag{5B.7ab}$$ The treatment given in the reference [I] provides all the results. Alternatively, we have seen in §3A that for a surface in conformal coordinates, the differential forms are $ω^μ=\d\xi^μ.$ The general induction formula is then $$\d\boldsymbol X=\d\xi^μ\boldsymbol E_μ,\tag{5B.8}$$ that is $$\boldsymbol X=x^i E_i =\boldsymbol X_0+\int_γ(\boldsymbol E_1\d\xi^1+\boldsymbol E_2\d\xi^2).\tag{5B.9}$$ On introducing the step numbers $$E_\pm=E_1\pm iE_2=σ_3⊗ iσ_\pm,\quad σ_\pm=σ_1\pm iσ_2,\tag{5B.10}$$ it becomes $$\hspace-19pt\boldsymbol X =\tfrac12(x^1+ix^2)E_-+\tfrac12(x^1-ix^2)E_++x^3E_3 =\boldsymbol X_0+\tfrac12\int_γ(\boldsymbol E_-\dζ+\boldsymbol E_+\d\barζ)\tag{5B.11}$$ with $$\boldsymbol E_\pm:=\tildeΨ E_\pmΨ=σ_3⊗\Phi^† iσ_\pm\Phi.\tag{5B.12}$$ Using the orthonormality identities $$\langle E^\pm E_\mp\rangle=-2E,\quad \langle E^\pm E_\pm\rangle=0,\quad \langle E^\pm E_3\rangle=0,\quad \langle E^3E_3\rangle=-E\tag{5B.13}$$ we get the components $$\hspace-4em\displaylines{x^1\pm ix^2 =x^1_0+ix^2_0-\tfrac12\langle\int_γ(σ^3⊗ iσ^\pm) (σ^3⊗\Phi^† iσ_-\Phi\dζ' +σ^3⊗\Phi^† iσ_+\Phi\d\barζ')\rangle\\ =x^1_0+ix^2_0+\int_γ(\tfrac14\bTr{σ^\pm\Phi^†σ_-\Phi}\dζ' +\tfrac14\bTr{σ^\pm\Phi^†σ_+\Phi}\d\barζ'),}\tag{5B.14a}$$ $$x^3 =x^3_0-\tfrac12\langle\int_γ(σ^3⊗ iσ^3)(σ^3⊗\Phi^† iσ_-\Phi\dζ' +σ^3⊗\Phi^† iσ_+\Phi\d\barζ')\rangle\\ =x^3_0+\int_γ(\tfrac14\bTr{σ^3\Phi^† σ_-\Phi}\dζ' +\tfrac14\bTr{σ^3\Phi^† σ_+\Phi}\d\barζ').\tag{5B.14b}$$ With the help of the general expressions $$\bTr{σ^+\Phi^†_ασ_-\Phi_β} =\phantom-4φ_αφ_β,\quad \bTr{σ^+\Phi^†_ασ_+\Phi_β} =-4χ_αχ_β,\tag{5B.15a}$$ $$\bTr{σ^-\Phi^†_ασ_-\Phi_β} =-4\barχ_α\barχ_β,\quad \bTr{σ^-\Phi^†_ασ_+\Phi_β} =\phantom-4\barφ_α\barφ_β,\tag{5B.15b}$$ $$\hspace-1em\bTr{σ^3\Phi^†_ασ_-\Phi_β} =2\barχ_αφ_β+2φ_α\barχ_β,\quad \bTr{σ^3\Phi^†_ασ_+\Phi_β} =2\barφ_αχ_β+2χ_α\barφ_β,\tag{5B.15c}$$ $$\bTr{\Phi^†_ασ_-\Phi_β} =2\barχ_αφ_β-2φ_α\barχ_β,\quad \bTr{\Phi^†_ασ_+\Phi_β} =2\barφ_αχ_β-2χ_α\barφ_β,\tag{5B.15d}$$ it gives the known formula [K96, I].

Space-like (Riemannian) surface in $\Bbb R^{2,1}$

For the representation of ${\rm C}\ell_{2,1},$ complex $2\times2$ matrices are enough. We first define $$\varsigma_1=σ_1,\quad\varsigma_2=σ_2,\quad\varsigma_3=iσ_3\tag{5B.16}$$ satisfying $$\varsigma_j\varsigma_k=-\varsigma_i^†,\tag{5B.17a}$$ $$\varsigma_1\varsigma_1=\varsigma_2\varsigma_2=-\varsigma_3\varsigma_3=\boldsymbol1,\tag{5B.17b}$$ and we choose $$E=\boldsymbol1,\quad E_i=i\varsigma_i,\quad E_{jk}=\varsigma^†_i,\quad E_{123}=i\boldsymbol1,\tag{5B.18}$$ satisfying $$-E_1E_1=-E_2E_2=+E_3E_3=E.\tag{5B.19}$$ The spinor is given by $$Ψ=ψ E+ψ^{jk}E_{jk}=:\Xi,\tag{5B.20}$$ $$\tildeΨ=\Xi^+,\tag{5B.$\tilde{20}$}$$ with $$\Xi=\pmatrix{φ& \barχ\\χ& \barφ}\in\Bbb R\times{\rm SU}(1,1),\quad \Xi^+=\pmatrix{\phantom-\barφ& -\barχ\\-χ& \phantom-φ}.\tag{5B.21}$$ The Dirac equation (5B.2) is represented as $$\{σ^2∂_1-σ^1∂_2-m\boldsymbol1\}\Xi=\boldsymbol0\tag{5B.22}$$ and splits into $$\left\{\eqalign{ \bar∂φ=-\tfrac12imχ\\ ∂χ=\phantom-\tfrac12imφ }\right.\tag{5B.23}$$ The step numbers and the tangent vectors are $$E_\pm=iσ_\pm,\tag{5B.24}$$ $$\boldsymbol E_\pm=\Xi^+ iσ_\pm\Xi.\tag{5B.25}$$ Using the formula (5B.11) and the orthonormality identities (5B.13) above, we get the components $$x^1\pm ix^2 =x^1_0\pm ix^2_0-\tfrac12\langle\int_γ iσ^\pm (\Xi^+ iσ_-\Xi\dζ' +\Xi^+ iσ_+\Xi\d\barζ')\rangle\\ =x^1_0\pm ix^2_0+\int_γ(\tfrac14\bTr{σ^\pm\Xi^+σ_-\Xi}\dζ' +\tfrac14\bTr{σ^\pm\Xi^+σ_+\Xi}\d\barζ'),\tag{5B.26a}$$ $$x^3 =x^3_0+\tfrac12\langle\int_γσ^3(\Xi^+ iσ_-\Xi\dζ +\Xi^† iσ_+\Xi\d\barζ)\rangle\\ =x^3_0+i\int_γ(\tfrac14\bTr{σ^3\Xi^+σ_-\Xi}\dζ +\tfrac14\bTr{σ^3\Xi^+σ_+\Xi}\d\barζ).\tag{5B.26b}$$ With the help of the general expressions $$\bTr{σ^+\Xi_α^+σ_-\Xi_β} =\phantom-4φ_αφ_β,\quad \bTr{σ^+\Xi_α^+σ_+\Xi_β} =-4χ_αχ_β,\tag{5B.27a}$$ $$\bTr{σ^-\Xi_α^+σ_-\Xi_β} =-4\barχ_α\barχ_β,\quad \bTr{σ^-\Xi_α^+σ_+\Xi_β} =\phantom-4\barφ_α\barφ_β,\tag{5B.27b}$$ $$\hspace-22pt\bTr{σ^3\Xi_α^+σ_-\Xi_β} =-2\barχ_αφ_β-2φ_α\barχ_β,\quad \bTr{σ^3\Xi_α^+σ_+\Xi_β} =\phantom-2\barφ_αχ_β+2χ_α\barφ_β,\tag{5B.27c}$$ $$\hspace-1em\bTr{\Xi_α^+σ_-\Xi_β} =-2\barχ_αφ_β+2φ_α\barχ_β,\quad \bTr{\Xi_α^+σ_+\Xi_β} =\phantom-2\barφ_αχ_β-2χ_α\barφ_β,\tag{5B.27d}$$ it gives the final formula [K96, CT99] $$x^1+ix^2 =x^1_0+ix^2_0+\int_γ(φφ~\dζ' -χχ~\d\barζ'),\tag{5B.28a}$$ $$x^1-ix^2 =x^1_0-ix^2_0+\int_γ(-\barχ\barχ~\dζ' +\barφ\barφ~\d\barζ'),\tag{5B.28b}$$ $$x^3 =x^3_0+i\int_γ(-φ\barχ~\dζ' +\barφχ~\d\barζ').\tag{5B.28c}$$

Time-like (Lorentzian) surface in $\Bbb R^{2,1}$ (motion of a plane curve)

We define the real matrices $${\sf s}_1=σ_1,\quad{\sf s}_2=iσ_2,\quad{\sf s}_3=σ_3\tag{5B.29}$$ satisfying $${\sf s}_i{\sf s}_j=-{\sf s}_k^t,\tag{5B.30a}$$ $$+{\sf s}_1{\sf s}_1=-{\sf s}_2{\sf s}_2=+{\sf s}_3{\sf s}_3=\boldsymbol1.\tag{5B.30b}$$ Another representation of ${\rm C}\ell_{2,1}$ is $$E=\boldsymbol1,\quad E_i=i{\sf s}_i,\quad E_{jk}={\sf s}^t_i,\quad E_{123}=i\boldsymbol1\tag{5B.31}$$ satisfying $$-E_1E_1=+E_2E_2=-E_3E_3=E.\tag{5B.32}$$ The spinor is given by $$Ψ=ψ E+ψ^{jk}E_{jk}=:Υ.\tag{5B.33}$$ $$\tildeΨ=Υ^\times\tag{5B.$\tilde{33}$}$$ with $$Υ=\pmatrix{a& b\\c& d}\in{\rm GL}(2\Bbb R),\quad Υ^\times:=\pmatrix{\phantom-d& -b\\-c& \phantom-a}.\tag{5B.34}$$ The Dirac equation (5B.2) is represented as $$\{-iσ^2∂_1+σ^1∂_2-m\boldsymbol1\}Υ=\boldsymbol0.\tag{5B.35}$$ Since the matrix elements of $Υ$ are real, we can use the variables $$φ=a+ib,\quadχ=c+id.\tag{5B.36}$$ In addition, we have to define another set of characteristic variables and derivatives $$ζ^-=\xi^1-\xi^2,\quadζ^+=\xi^1+\xi^2,\tag{5B.37}$$ $$∂_-:=∂_{ζ^-}=\tfrac12(∂_1-∂_2),\quad ∂_+:=∂_{ζ^+}=\tfrac12(∂_1+∂_2).\tag{5B.38}$$ Then the equation splits into $$\left\{\eqalign{ ∂_-φ& =\phantom-\tfrac12mχ,\\ ∂_+χ& =-\tfrac12mφ. }\right.\tag{5B.39ab}$$ We also introduce other step numbers. $$E'_\pm=E_1\pm E_2=iσ_\pm,\quad (E'^\pm=E^1\mp E^2=iσ^\mp).\tag{5B.40}$$ The orthonormality identities are similar : $$\langle E'^\pm E'_\mp\rangle=-2E,\quad \langle E'^\pm E'_\pm\rangle=0,\quad \langle E'^\pm E_3\rangle=0,\quad \langle E^3E_3\rangle=-E,\tag{5B.41}$$ and the induction formula is $$\hspace-33pt\boldsymbol X =\tfrac12(x^1-x^2)E'_-+\tfrac12(x^1+x^2)E'_++x^3E_3 =\boldsymbol X_0 +\tfrac12\int_γ(\boldsymbol E'_-\dζ'^-+\boldsymbol E'_+\dζ'^+),\tag{5B.42}$$ with $$\boldsymbol E'_\pm=Υ^\times iσ_\pmΥ.\tag{5B.43}$$ We get the components $$x^1\pm x^2 =x^1_0\pm x^2_0-\tfrac12\langle\int_γ iσ^\mp [Υ^\times iσ_-Υ\dζ'^- +Υ^\times iσ_+Υ\dζ'^+]\rangle\\ =x^1_0\pm x^2_0+\int_γ(\tfrac14\bTr{σ^\mpΥ^\timesσ_-Υ}\dζ'^- +\tfrac14\bTr{σ^\mpΥ^\timesσ_+Υ}\dζ'^+),\tag{5B.44a}$$ $$x^3 =x^3_0-\tfrac12\langle\int_γ iσ^3[ Υ^\times iσ_-Υ\dζ'^- +Υ^\times iσ_+Υ\dζ'^+]\rangle\\ =x^3_0+\int_γ( \tfrac14\bTr{σ^3Υ^\timesσ_-Υ}\dζ'^- +\tfrac14\bTr{σ^3Υ^\timesσ_+Υ}\dζ'^+).\tag{5B.44b}$$ With the help of the general expressions $$\bTr{σ^+Υ^\times_ασ_-Υ_β} =\phantom-4a_α a_β,\quad \bTr{σ^+Υ^\times_ασ_+Υ_β} =-4c_α c_β,\tag{5B.45a}$$ $$\bTr{σ^-Υ^\times_ασ_-Υ_β} =-4b_α b_β,\quad \bTr{σ^-Υ^\times_ασ_+Υ_β} =\phantom-4d_α d_β,\tag{5B.45b}$$ $$\hspace-19pt\bTr{σ^3Υ^\times_ασ_-Υ_β} =-2a_α b_β-2b_α a_β,\quad \bTr{σ^3Υ^\times_ασ_+Υ_β} =\phantom-2c_α d_α+2d_α c_β,\tag{5B.45c}$$ $$\bTr{Υ^\times_ασ_-Υ_β} =\phantom-2a_α b_β-2b_α a_β,\quad \bTr{Υ^\times_ασ_+Υ_β} =\phantom-2d_α c_β-2c_α d_β,\tag{5B.45d}$$ it gives the final formula [L’08, P’16] $$x^1+x^2 =x^1_0+x^2_0+\int_γ(-b^2~\dζ'^-+d^2~\dζ'^+),\tag{5B.46a}$$ $$x^1-x^2 =x^1_0-x^2_0+\int_γ(a^2~\dζ'^--c^2~\dζ'^+),\tag{5B.46b}$$ $$x^3 =x^3_0+\int_γ(-ab~\dζ'^+-cd~\dζ'^+).\tag{5B.46c}$$

Four dimensions

For $p=2$ and $q=2,$ It is always possible by a gauge transformation to get rid of the potential term, that plays no part in the description of the surface. We are left with the Dirac equation $$\{E^1∂_1+E^2∂_2+m_3E^3+m_4E^4\}Ψ=\boldsymbol0\tag{5B.47}$$ which multiplied on the left by $E_3$ is $$\{E^{\ph1}_3∂_1+E^{\ph2}_3∂_2-m_3+m_4E^{\ph4}_3\}Ψ=\boldsymbol0\tag{5B.48}$$

Surface in $\Bbb R^{4,0}$

We take as a representation of ${\rm C}\ell_{4,0}$ : $$E=\boldsymbol1⊗\boldsymbol1,\quad \eqalign{E_i& =iσ_2⊗σ_i\\E_4& =σ_1⊗ i\boldsymbol1},\quad \eqalign{E_{jk}& =-\boldsymbol1⊗ iσ_i\\E_{4i}& =-σ_3⊗ iσ_i},\\ \eqalign{E_{4jk}& =σ_1⊗σ_i\\E_{123}& =σ_2⊗\boldsymbol1},\quad E_{1234}=σ_3⊗\boldsymbol1.\tag{5B.49}$$ The spinor is given by $$\eqalign{ Ψ& =(ψ E+ψ^{jk}E_{jk})+(ψ^{1234}E_{1234}+ψ^{4i}E_{4i})\\ & =\boldsymbol1⊗(ψ\boldsymbol1-iψ^{jk}σ_i) +σ_3⊗(ψ^{1234}\boldsymbol1-iψ^{4i}σ_i)\\ & =:\boldsymbol1⊗\tfrac12(\Phi_1+\Phi_2)+σ_3⊗\tfrac12(\Phi_1-\Phi_2),}\tag{5B.50}$$ $$\tildeΨ =\boldsymbol1⊗\tfrac12(\Phi_1^†+\Phi_2^†) +σ_3⊗\tfrac12(\Phi_1^†-\Phi_2^†),\tag{5B.$\tilde{50}$}$$ where $$\Phi_1,\Phi_2\in\Bbb R\times{\rm SU}(2).\tag{5B.51}$$ It is not really an element of the special Clifford group unless $$\Phi_1^†\Phi_1=\Phi_2^†\Phi_2,\tag{5B.52}$$ but the vectors are still transformed into vectors, so that it amounts to the same and is simpler. The Dirac equation (5B.48) is represented as $$\begin{multline}\hspace{6em} \{-\boldsymbol1⊗ iσ^2∂_1 +\boldsymbol1⊗ iσ^1∂_2 -m_3\boldsymbol1⊗\boldsymbol1 +m_4σ^3⊗ iσ^3\}\times\\ [\boldsymbol1⊗\tfrac12(\Phi_1+\Phi_2) +σ_3⊗\tfrac12(\Phi_1-\Phi_2)]=0, \hspace{6em}\tag{5B.53}\end{multline}$$ and splits into $$\{-iσ^2∂_1+iσ^1∂_2-m_3\boldsymbol1+m_4iσ^3\}\Phi_1=\boldsymbol0,\\ \{-iσ^2∂_1+iσ^1∂_2-m_3\boldsymbol1-m_4iσ^3\}\Phi_2=\boldsymbol0.\tag{5B.54ab}$$ It splits further to $$\left\{\eqalign{ \bar∂φ_1=\phantom-\tfrac12μχ_1& ,\quad \bar∂φ_2=\phantom-\tfrac12\barμχ_2~,\\ ∂χ_1=-\tfrac12\barμφ_1& ,\quad ∂χ_2=-\tfrac12μφ_2~. }\right.\tag{5B.55abcd}$$ with $$μ:=m_3+im_4.\tag{5B.56}$$ The step numbers and the tangent vectors are $$E_\pm=E_1\pm iE_2=σ_2⊗ iσ_\pm,\tag{5B.57}$$ $$\boldsymbol E_\pm =σ_+⊗\tfrac12\Phi_1^†σ_\pm\Phi_2 -σ_-⊗\tfrac12\Phi_2^†σ_\pm\Phi_1.\tag{5B.58}$$ With the orthonormality identities $$\langle E^\pm E_\mp\rangle=-2E,\quad \langle E^\pm E_\pm\rangle=0,\quad \langle E^\pm E_3\rangle=0,\quad \langle E^\pm E_4\rangle=0,\\ \langle E^3E_3\rangle=-E,\quad \langle E^3E_4\rangle=0,\quad \langle E^4E_4\rangle=-E,\tag{5B.59}$$ and the induction formula $$\eqalign{ \boldsymbol X & =\tfrac12(x^1+ix^2)E_-+\tfrac12(x^1-ix^2)E_++x^3E_3+x^4E_4\\ & =\boldsymbol X_0+\tfrac12\int_γ(\boldsymbol E_-\dζ +\boldsymbol E_+\d\barζ),}\tag{5B.60}$$ we get the components $$\hspace-5em\displaylines{ x^1\pm ix^2 =x^1_0\pm ix^2_0 -\tfrac12\langle\int_γ(σ^2⊗ iσ^\pm) [(σ_+⊗\tfrac12\Phi_1^†σ_-\Phi_2-σ_ -⊗\tfrac12\Phi_2^†σ_-\Phi_1)\dζ'+\\ +(σ_+⊗\tfrac12\Phi_1^†σ_+\Phi_2-σ_ -⊗\tfrac12\Phi_2^†σ_+\Phi_1)\d\barζ']\rangle\\ =x^1_0\pm ix^2_0 +\int_γ(\tfrac18\bTr{σ^\pm\Phi_1^†σ_-\Phi_2 +σ^\pm\Phi_2^†σ_-\Phi_1}\dζ'+\\ +\tfrac18\bTr{σ^\pm\Phi_1^†σ_+\Phi_2 +σ^\pm\Phi_2^†σ_+\Phi_1}\d\barζ'),}\tag{5B.61a}$$ $$x^3=x^3_0-\tfrac12\langle\int_γ(σ^2⊗ iσ^3)[ (σ_+⊗\tfrac12\Phi_1^†σ_-\Phi_2 -σ_-⊗\tfrac12\Phi_2^†σ_-\Phi_1)\dζ'+\\ +(σ_+⊗\tfrac12\Phi_1^†σ_+\Phi_2 -σ_-⊗\tfrac12\Phi_2^†σ_+\Phi_1)\d\barζ']\rangle\\ =x^3_0+\int_γ(\tfrac18\bTr{σ^3\Phi_1^†σ_-\Phi_2 +σ^3\Phi_2^†σ_-\Phi_1}\dζ'+\\ +\tfrac18\bTr{σ^3\Phi_1^†σ_+\Phi_2 +σ^3\Phi_2^†σ_+\Phi_1}\d\barζ'),\tag{5B.61b}$$ $$\hspace-3em\displaylines{x^4=x^4_0-\tfrac12\langle\int_γ(σ^1⊗ i\boldsymbol1) [(σ_+⊗\tfrac12\Phi_1^†σ_+\Phi_2 -σ_-⊗\tfrac12\Phi_2^†σ_+\Phi_1)\dζ'+\\ +(σ_+⊗\tfrac12\Phi_1^†σ_-\Phi_2 -σ_-⊗\tfrac12\Phi_2^†σ_-\Phi_1)\d\barζ']\rangle\\ =x^4_0-i\int_γ( \tfrac18\bTr{\Phi_1^†σ_+\Phi_2 -\Phi_2^†σ_+\Phi_1}\dζ' +\tfrac18\bTr{\Phi_1^†σ_-\Phi_2 -\Phi_2^†σ_-\Phi_1}\d\barζ').}\tag{5B.61c}$$ With the help of the general expressions (5B.15) it gives the final formula [K96] $$x^1+ix^2=x^1_0+ix^2_0+\int_γ(φ_1φ_2~\dζ' -χ_1χ_2~\d\barζ'),\tag{5B.62a}$$ $$x^1-ix^2=x^1_0-ix^2_0+\int_γ(-\barχ_1\barχ_2~\dζ' +\barφ_1\barφ_2~\d\barζ')$$ $$x^3+ix^4=x^3_0+ix^4_0+\int_γ(\barχ_1φ_2~\dζ' +\barφ_1χ_2~\d\barζ'),\tag{5B.62b}$$ $$x^3-ix^4=x^3_0-ix^4_0+\int_γ(φ_1\barχ_2~\dζ' +χ_1\barφ_2~\d\barζ').\tag{5B.62c}$$

Space-like (Riemannian) surface in $\Bbb R^{2,2}$

Using the symbols (5B.16), a representation of ${\rm C}\ell_{2,2}$ is $$E=\boldsymbol1⊗\boldsymbol1,\quad \eqalign{E_i& =iσ_2⊗\varsigma_i\\ E_4& =\phantom iσ_1⊗\boldsymbol1},\quad \eqalign{E_{jk}& =\phantom-\boldsymbol1⊗\varsigma_i^†\\ E_{4i}& =-σ_3⊗\varsigma_i},\\ \eqalign{E_{4jk}& =σ_1⊗\varsigma_i^†\\ E_{123}& =σ_2⊗ i\boldsymbol1},\quad E_{1234}=σ_3⊗\boldsymbol1,\tag{5B.63}$$ satisfying $$-E_1E_1=-E_2E_2=+E_3E_3=+E_4E_4=E.\tag{5B.64}$$ The spinor is given by $$\eqalign{Ψ& =ψ E+ψ^{jk}E_{jk}+ψ^{4i}E_{4i}+ψ^{1234}E_{1234}\\ & =\boldsymbol1⊗(ψ\boldsymbol1+ψ^{jk}\varsigma_i^†) +σ_3⊗(ψ^{1234}\boldsymbol1-ψ^{4i}\varsigma_i)\\ & =:\boldsymbol1⊗\tfrac12(\Xi_1+\Xi_2) +σ_3⊗\tfrac12(\Xi_1-\Xi_2),}\tag{5B.65}$$ $$\eqalign{\tildeΨ& =ψ E-ψ^{jk}E_{jk}-ψ^{4i}E_{4i}+ψ^{1234}E_{1234}\\ & =:\boldsymbol1⊗\tfrac12(\Xi_1^++\Xi_2^+) +σ_3⊗\tfrac12(\Xi_1^+-\Xi_2^+),}\tag{5B.$\tilde{65}$}$$ and $\Xi\in\Bbb R\times{\rm SU}(1,1)$ is defined by (5B.21). The Dirac equation (5B.48) is represented as $$\begin{multline}\hspace{6em} \{\boldsymbol1⊗σ^2∂_1 -\boldsymbol1⊗σ^1∂_2 -m_3\boldsymbol1⊗\boldsymbol1-m_4σ_3⊗ iσ^3\}\times\\ [\boldsymbol1⊗\tfrac12(\Xi_1+\Xi_2) +σ_3⊗\tfrac12(\Xi_1-\Xi_2)]=\boldsymbol0 \hspace{6em}\tag{5B.66}\end{multline}$$ and splits into $$\{σ^2∂_1-σ^1∂_2-m_3\boldsymbol1-m_4iσ^3\}\Xi_1\\ \{σ^2∂_1-σ^1∂_2-m_3\boldsymbol1+m_4iσ^3\}\Xi_2.\tag{5B.67ab}$$ It splits further in $$\left\{\eqalign{ \bar∂φ_1=-\tfrac12i\barμχ_1\quad \bar∂φ_2=-\tfrac12iμχ_2,\\ ∂χ_1=\phantom-\tfrac12iμφ_1\quad ∂χ_2=\phantom-\tfrac12i\barμφ_2. }\right.\tag{5B.68abcd}$$ The step numbers and the tangent vectors are $$E_\pm=E_1\pm iE_2=σ_2⊗ iσ_\pm,\tag{5B.69}$$ $$\boldsymbol E_\pm =σ_+⊗\tfrac12\Xi^+_1σ_\pm\Xi_2 -σ_-⊗\tfrac12\Xi^+_2σ_\pm\Xi_1.\tag{5B.70}$$ From the induction formula (5B.60) and the orthonormality identities (5B.59) we get the components $$\hspace-38pt\displaylines{x^1\pm ix^2 =x^1_0\pm ix^2_0-\tfrac12\langle\int_γ (σ^2⊗ iσ^\pm)[(σ_+⊗\tfrac12\Xi^+_1σ_-\Xi_2 -σ_-⊗\tfrac12\Xi^+_2σ_-\Xi_1)\dζ'+\\ +(σ_+⊗\tfrac12\Xi^+_1σ_+\Xi_2 -σ_-⊗\tfrac12\Xi^+_2σ_+\Xi_1)\d\barζ']\rangle\\ =x^1_0\pm ix^2_0+\int_γ (\tfrac18\bTr{σ^\pm\Xi^+_1σ_-\Xi_2+σ^\pm\Xi^+_2σ_-\Xi_1}\dζ'+\\ +\tfrac18\bTr{σ^\pm\Xi^+_1σ_+\Xi_2+σ^\pm\Xi^+_2σ_+\Xi_1}\d\barζ'),}\tag{5B.71a}$$ $$x^3 =x^3_0-\tfrac12\langle\int_γ (σ^2⊗σ^3)[ (σ_+⊗\tfrac12\Xi^+_1σ_-\Xi_2 -σ_-⊗\tfrac12\Xi^+_2σ_-\Xi_1)\dζ'+\\ +(σ_+⊗\tfrac12\Xi^+_1σ_+\Xi_2 -σ_-⊗\tfrac12\Xi^+_2σ_+\Xi_1)\d\barζ']\rangle\\ =x^3_0+i\int_γ (\tfrac18\bTr{σ^3\Xi^+_1σ_-\Xi_2+σ^3\Xi^+_2σ_-\Xi_1}\dζ'+\\ +\tfrac18\bTr{σ^3\Xi^+_1σ_+\Xi_2+σ^3\Xi^+_2σ_+\Xi_1}\d\barζ'),\tag{5B.71b}$$ $$\hspace-18pt\displaylines{x^4 =x^4_0-\tfrac12\langle\int_γ (-σ^1⊗\boldsymbol1)[ (σ_+⊗\tfrac12\Xi^+_1σ_-\Xi_2 -σ_-⊗\tfrac12\Xi^+_2σ_-\Xi_1)\dζ'+\\ +(σ_+⊗\tfrac12\Xi^+_1σ_+\Xi_2 -σ_-⊗\tfrac12\Xi^+_2σ_+\Xi_1)\d\barζ']\rangle\\ =x^4_0+\int_γ (\tfrac18\bTr{\Xi^+_1σ_-\Xi_2-\Xi^+_2σ_-\Xi_1}\dζ' +\tfrac18\bTr{\Xi^+_1σ_+\Xi_2-\Xi^+_2σ_+\Xi_1}\d\barζ').}\tag{5B.71c}$$ With the help of the general expressions (5B.27) it gives the final formula [K96] $$x^1+ix^2 =x^1_0+ix^2_0+\int_γ (φ_1φ_2~\dζ' -χ_1χ_2~\d\barζ'),\tag{5B.72a}$$ $$x^1-ix^2 =x^1_0-ix^2_0+\int_γ (-\barχ_1\barχ_2\dζ' +\barφ_1\barφ_2\d\barζ'),\tag{5B.72b}$$ $$x^3+ix^4 =x^3_0+ix^4_0+i\int_γ (-\barχ_1φ_2~\dζ' +\barφ_1χ_2~\d\barζ'),\tag{5B.72c}$$ $$x^3-ix^4 =x^3_0-ix^4_0+i\int_γ (-φ_1\barχ_2~\dζ' +χ_1\barφ_2~\d\barζ').\tag{5B.72d}$$

Time-like (Lorentzian) surface in $\Bbb R^{2,2}$

Using the symbols (5B.29), another representation of ${\rm C}\ell_{2,2}$ is $$E=\boldsymbol1⊗\boldsymbol1,\quad \eqalign{E_i& ={\sf s}_2⊗{\sf s}_i\\ E_4& ={\sf s}_1⊗\boldsymbol1},\quad \eqalign{E_{jk}& =\phantom-\boldsymbol1⊗{\sf s}_i^t\\ E_{4i}& =-{\sf s}_3⊗{\sf s}_i},\\ \eqalign{E_{4jk}& ={\sf s}_1⊗{\sf s}_i^t\\ E_{123}& ={\sf s}_2⊗\boldsymbol1},\quad E_{1234}={\sf s}_3⊗\boldsymbol1.\tag{5B.73}$$ satisfying $$-E_1E_1=+E_2E_2=-E_3E_3=+E_4E_4=E.\tag{5B.74}$$ All matrices are real. The spinor is given by $$\eqalign{Ψ& =ψ E+ψ^{jk}E_{jk}+ψ^{4i}E_{4i}+ψ^{1234}E_{1234}\\ & =\boldsymbol1⊗(ψ\boldsymbol1+ψ^{jk}{\sf s}^t_i) +σ_3⊗(ψ^{1234}\boldsymbol1-ψ^{4i}{\sf s}_i)\\ & =:\boldsymbol1⊗\tfrac12(Υ_1+Υ_2) +σ_3⊗\tfrac12(Υ_1-Υ_2),}\tag{5B.75}$$ $$\eqalign{\tildeΨ& =ψ E-ψ^{jk}E_{jk}-ψ^{4i}E_{4i}+ψ^{1234}E_{1234}\\ & =:\boldsymbol1⊗\tfrac12(Υ_1^\times+Υ_2^\times) +σ_3⊗\tfrac12(Υ_1^\times-Υ_2^\times),}\tag{5B.$\tilde{75}$}$$ where $Υ\in{\rm GL}(2\Bbb R)$ is defined in (5B.34). The Dirac equation (5B.48) is represented as $$\{-\boldsymbol1⊗iσ^2∂_1+\boldsymbol1⊗σ^1∂_2 -m_3\boldsymbol1⊗\boldsymbol1-m_4σ^3⊗σ^3\}Ψ=\boldsymbol0.\tag{5B.76}$$ It splits into $$\{-iσ^2∂_1+σ^1∂_2 -m_3\boldsymbol1-m_4σ^3\}Υ_1=\boldsymbol0,\\ \{-iσ^2∂_1+σ^1∂_2 -m_3\boldsymbol1+m_4σ^3\}Υ_2=\boldsymbol0,\tag{5B.77ab}$$ and further in $$\left\{\eqalign{ ∂_+φ_1=\phantom-\tfrac12m_-χ_1,\quad ∂_+φ_2=\phantom-\tfrac12m_+χ_2,\\ ∂_-χ_1=-\tfrac12m_+φ_1,\quad ∂_-χ_2=-\tfrac12m_-φ_2. }\right.\tag{5B.78abcd}$$ The step numbers and the tangent vectors are $$E'_\pm=E_1\pm E_2=iσ_2⊗σ_\pm\quad(E'^\pm=E^1\mp E^2=iσ^2⊗σ^\mp),\tag{5B.79}$$ $$\boldsymbol E'_\pm =σ_+⊗\tfrac12Υ_1^\timesσ_\pmΥ_2 -σ_-⊗\tfrac12Υ_2^\timesσ_\pmΥ_1.\tag{5B.80}$$ From the induction formula $$\eqalign{ \boldsymbol X & =\tfrac12(x^1-x^2)E'_-+\tfrac12(x^1+x^2)E'_++x^3E_3+x^4E_4\\ & =\boldsymbol X_0 +\tfrac12\int_γ(\boldsymbol E'_-\dζ'^-+\boldsymbol E'_+\dζ'^+) }\tag{5B.81}$$ and the orthonormality identities (5B.59), we get the components $$\hspace-41pt\displaylines{x^1\pm x^2 =x^1_0\pm x^2_0-\tfrac12\langle\int_γ (iσ^2⊗σ^\mp) [(σ_+⊗\tfrac12Υ_1^\timesσ_-Υ_2 -σ_-⊗\tfrac12Υ_2^\timesσ_-Υ_1)\dζ'^-+\\ +(σ_+⊗\tfrac12Υ_1^\timesσ_+Υ_2 -σ_-⊗\tfrac12Υ_2^\timesσ_+Υ_1)\dζ'^+]\rangle\\ =x^1_0\pm x^2_0+\int_γ (\tfrac18\bTr{σ^\mpΥ_1^\timesσ_-Υ_2 +σ^\mpΥ_2^\timesσ_-Υ_1}\dζ'^-+\\ +\tfrac18\bTr{σ^\mpΥ_1^\timesσ_+Υ_2 +σ^\mpΥ_2^\timesσ_+Υ_1}\dζ'^+),}\tag{5B.82a}$$ $$x^3 =x^3_0-\tfrac12\langle\int_γ(iσ^2⊗σ^3) [(σ_+⊗\tfrac12Υ_1^\timesσ_-Υ_2 -σ_-⊗\tfrac12Υ_2^\timesσ_-Υ_1)\dζ'^-+\\ +(σ_+⊗\tfrac12Υ_1^\timesσ_+Υ_2 -σ_-⊗\tfrac12Υ_2^\timesσ_+Υ_1)\dζ'^+]\rangle\\ =x^3_0+\int_γ(\tfrac18\bTr{σ^3Υ_1^\timesσ_-Υ_2 +σ^3Υ_2^\timesσ_-Υ_1}\dζ'^-+\\ +\tfrac18\bTr{σ^3Υ_1^\timesσ_+Υ_2 +σ^3Υ_2^\timesσ_+Υ_1}\dζ'^+),\tag{5B.82b}$$ $$\hspace-48pt\displaylines{x^4 =x^4_0-\tfrac12\int_γ(-σ_1⊗\boldsymbol1) ([σ_+⊗\tfrac12Υ_1^\timesσ_-Υ_2 -σ_-⊗\tfrac12Υ_2^\timesσ_-Υ_1]\dζ'^-+\\ +[σ_+⊗\tfrac12Υ_1^\timesσ_+Υ_2 -σ_-⊗\tfrac12Υ_2^\timesσ_+Υ_1]\dζ'^+).\\ =x^4_0+\tfrac12\int_γ (\tfrac18\bTr{Υ_1^\timesσ_-Υ_2 -Υ_2^\timesσ_-Υ_1}\dζ'^- +\tfrac18\bTr{Υ_1^\timesσ_+Υ_2 -Υ_2^\timesσ_+Υ_1}\dζ'^+).}\tag{5B.82c}$$ With the help of the general expressions (5B.45), it gives the final formula [K96, P’16] $$x^1+x^2 =x^1_0+x^2_0+\int_γ(-b_1b_2~\dζ'^-+d_1d_2~\dζ'^+),\tag{5B.83a}$$ $$x^1-x^2 =x^1_0-x^2_0+\int_γ(a_1a_2~\dζ'^--c_1c_2~\dζ'^+),\tag{5B.83b}$$ $$x^3+x^4=x^3_0+x^4_0+\int_γ(-b_1a_2~\dζ'^-+d_1c_2~\dζ'^+),\tag{5B.83c}$$ $$x^3-x^4=x^3_0-x^4_0+\int_γ(-a_1b_2~\dζ'^-+c_1d_2~\dζ'^+).\tag{5B.83d}$$

Space-like (Riemannian) surface in $\Bbb R^{3,1}$

A representation of ${\rm C}\ell_{3,1}$ is $$E=\boldsymbol1⊗\boldsymbol1,\quad \eqalign{E_i& =iσ_2⊗σ_i\\ E_4& =-σ_1⊗\boldsymbol1},\quad \eqalign{E_{jk}& =-\boldsymbol1⊗iσ_i\\ E_{4i}& =\phantom-σ_3⊗σ_i},\\ \eqalign{E_{4jk}& =σ_1⊗iσ_i\\ E_{123}& =σ_2⊗\boldsymbol1},\quad E_{1234}=iσ_3⊗\boldsymbol1,\tag{5B.84}$$ satisfying $$-E_1E_1=-E_2E_2=-E_3E_3=+E_4E_4=E.\tag{5B.85}$$ The spinor is given by $$Ψ=\boldsymbol1⊗(ψ-ψ^{jk}iσ_i)+iσ_3⊗(ψ^{1234}-ψ^{4i}iσ_i)\\ =:\boldsymbol1⊗\Phi'+iσ_3⊗\Phi'' =:\boldsymbol1⊗\tfrac12(Υ_{12}+Υ_{21}) +σ_3⊗\tfrac12(Υ_{12}-Υ_{21}),\tag{5B.86}$$ $$\tildeΨ=:\boldsymbol1⊗\Phi'^†+iσ_3⊗\Phi''^† =\boldsymbol1⊗\tfrac12(Υ^†_{12}+Υ^†_{21}) -σ_3⊗\tfrac12(Υ^†_{12}-Υ^†_{21}),\tag{5B.$\tilde{86}$}$$ where $Υ_{12},Υ_{21}\in{\rm GL}(2\Bbb C)$ are defined as $$Υ_{12}=\Phi'+i\Phi''=\pmatrix{φ'+iφ''& -\barχ'-i\barχ''\\ χ'+iχ''& \phantom-\barφ'+i\barφ''} =:\pmatrix{φ_1& -\barχ_2\\χ_1& \phantom-\barφ_2},\tag{5B.87a}$$ $$Υ_{21}=\Phi'-i\Phi''=\pmatrix{φ'-iφ''& -\barχ'+i\barχ''\\ χ'-iχ''& \phantom-\barφ'-i\barφ''} =\pmatrix{φ_2& -\barχ_1\\χ_2& \phantom-\barφ_1}.\tag{5B.87b}$$ The Dirac equation (5B.48) is represented as $$\begin{multline}\hspace{6em} \{-\boldsymbol1⊗ iσ^2∂_1 +\boldsymbol1⊗ iσ^1∂_2 -m_3\boldsymbol1⊗\boldsymbol1 -m_4σ^3⊗σ^3\}\times\\ [\boldsymbol1⊗\tfrac12(Υ_{12}+Υ_{21}) +σ_3⊗\tfrac12(Υ_{12}-Υ_{21})]=\boldsymbol0. \hspace{6em}\tag{5B.88}\end{multline}$$ It splits into $$\{-iσ^2∂_1+iσ^1∂_2 -m_3+m_4σ^3\}Υ_{12}=\boldsymbol0,\\ \{-iσ^2∂_1+iσ^1∂_2 -m_3-m_4σ^3\}Υ_{21}=\boldsymbol0,\tag{5B.89ab}$$ and reduces further in $$\left\{\eqalign{ \bar∂φ_1=\phantom-\tfrac12m_+χ_1,\quad \bar∂φ_2=\phantom-\tfrac12m_-χ_2,\\ ∂χ_1=-\tfrac12m_-φ_1,\quad ∂χ_2=-\tfrac12m_+φ_2, }\right.\tag{5B.90abcd}$$ and complex conjugates. The step numbers and the tangent vectors are $$E_\pm=E_1\pm iE_2=iσ_2⊗σ_\pm,\tag{5B.91}$$ $$\boldsymbol E_\pm =σ_+⊗\tfrac12Υ^†_{21}σ_\pmΥ_{21} -σ_-⊗\tfrac12Υ^†_{12}σ_\pmΥ_{12}.\tag{5B.92}$$ From the induction formula (5B.60) and the orthonormality identities (5B.59) we get the components $$\hspace-46pt\displaylines{x^1\pm ix^2 =x^1_0\pm ix^2_0-\tfrac12\langle\int_γ (iσ^2⊗σ^\pm) [(σ_+⊗\tfrac12Υ^†_{21}σ_-Υ_{21} -σ_-⊗\tfrac12Υ^†_{12}σ_-Υ_{12})\dζ'+\\ +(σ_+⊗\tfrac12Υ^†_{21}σ_+Υ_{21} -σ_-⊗\tfrac12Υ^†_{12}σ_+Υ_{12})\d\barζ']\rangle\\ =x^1_0\pm ix^2_0+\int_γ (\tfrac18\bTr{σ^\pmΥ^†_{12}σ_-Υ_{12} +σ^\pmΥ^†_{21}σ_-Υ_{21}}\dζ'+\\ +\tfrac18\bTr{σ^\pmΥ^†_{12}σ_+Υ_{12} +σ^\pmΥ^†_{21}σ_+Υ_{21}}\d\barζ'),}\tag{5B.93a}$$ $$\hspace-1em x^3 =x^3_0-\tfrac12\langle\int_γ (iσ^2⊗σ^3) [(σ_+⊗\tfrac12Υ^†_{21}σ_-Υ_{21} -σ_-⊗\tfrac12Υ^†_{12}σ_-Υ_{12})\dζ'+\\ +(σ_+⊗\tfrac12Υ^†_{21}σ_+Υ_{21} -σ_-⊗\tfrac12Υ^†_{12}σ_+Υ_{12})\d\barζ']\rangle\\ =x^3_0+\int_γ (\tfrac18\bTr{σ^3Υ^†_{12}σ_-Υ_{12} +σ^3Υ^†_{21}σ_-Υ_{21}}\dζ'+\\ +\tfrac18\bTr{σ^3Υ^†_{12}σ_+Υ_{12} +σ^3Υ^†_{21}σ_+Υ_{21}}\d\barζ'),\tag{5B.93b}$$ $$\hspace-49pt\displaylines{x^4 =x^4_0-\tfrac12\langle\int_γ (σ^1⊗\boldsymbol1) [(σ_+⊗\tfrac12Υ^†_{21}σ_-Υ_{21} -σ_-⊗\tfrac12Υ^†_{12}σ_-Υ_{12})\dζ'+\\ +(σ_+⊗\tfrac12Υ^†_{21}σ_+Υ_{21} -σ_-⊗\tfrac12Υ^†_{12}σ_+Υ_{12})\d\barζ']\rangle\\ =x^4_0+\int_γ (\tfrac18\bTr{Υ^†_{12}σ_-Υ_{12} -Υ^†_{21}σ_-Υ_{21}}\dζ' +\tfrac18\bTr{Υ^†_{12}σ_+Υ_{12} -Υ^†_{21}σ_+Υ_{21}}\d\barζ').}\tag{5B.93c}$$ The evaluation of the traces gives the final formula [K96] $$x^1+ix^2 =x^1_0+ix^2_0+\int_γ (φ_1φ_2~\dζ'-χ_1χ_2~\d\barζ'),\tag{5B.94a}$$ $$x^1-ix^2 =x^1_0-ix^2_0+\int_γ (-\barχ_1\barχ_2~\dζ'+\barφ_1\barφ_2\d\barζ'),\tag{5B.94b}$$ $$x^3+x^4 =x^3_0+x^4_0+\int_γ (\barχ_1φ_1~\dζ'+\barφ_1χ_1~\d\barζ'),\tag{5B.94c}$$ $$x^3-x^4 =x^3_0-x^4_0+\int_γ (φ_2\barχ_2~\dζ'+χ_2\barφ_2~\d\barζ').\tag{5B.94d}$$

Time-like (Lorentzian) surface in $\Bbb R^{3,1}$ (motion of a space curve)

Using the symbols defined in (5B.29), another representation of ${\rm C}\ell_{3,1}$ is $$E=\boldsymbol1⊗\boldsymbol1,\quad \eqalign{E_i& =iσ_2⊗{\sf s}_i\\ E_4& =\phantom iσ_1⊗ i\boldsymbol1},\quad \eqalign{E_{jk}& =\boldsymbol1⊗{\sf s}^t_i\\ E_{i4}& =σ_3⊗ i{\sf s}_i},\\ \eqalign{E_{jk4}& =\phantom iσ_1⊗ i{\sf s}^t_i\\ E_{123}& =iσ_2⊗\boldsymbol1},\quad E_{1234}=σ_3⊗ i\boldsymbol1\tag{5B.95}$$ satisfying $$-E_1E_1=+E_2E_2=-E_3E_3=-E_4E_4=E.\tag{5B.96}$$ The spinor is given by $$\eqalign{Ψ& =\boldsymbol1⊗(ψ\boldsymbol1+ψ^{jk}{\sf s}^t_i) +σ_3⊗i(ψ^{1234}\boldsymbol1-ψ^{4i}{\sf s}_i)\\ & =:\boldsymbol1⊗Υ'+σ_3⊗ iΥ'' =:\boldsymbol1⊗\tfrac12(Υ+\barΥ) +σ_3⊗\tfrac12(Υ-\barΥ),}\tag{5B.97}$$ $$\tildeΨ=\boldsymbol1⊗\tfrac12(Υ^\times+\barΥ^\times) +σ_3⊗\tfrac12(Υ^\times-\barΥ^\times),\tag{5B.$\tilde{97}$}$$ where $$Υ:=Υ'+iΥ''=\pmatrix{α& β\\γ& δ}\in{\rm GL}(2\Bbb C),\quad \barΥ:=Υ'-iΥ''=\pmatrix{\barα& \barβ\\\barγ& \barδ}\tag{5B.98}$$ is now complex. The Dirac equation (5B.48) is represented by $$\begin{multline}\hspace{6em} \{-\boldsymbol1⊗ iσ^2∂_1 +\boldsymbol1⊗σ^1∂_2 -m_3\boldsymbol1⊗\boldsymbol1 +m_4σ_3⊗ iσ^3\}\times\\ [\boldsymbol1⊗\tfrac12(Υ+\barΥ) +σ^3⊗\tfrac12(Υ-\barΥ)]=\boldsymbol0. \hspace{6em}\tag{5B.99}\end{multline}$$ It spits into $$\{-iσ^2∂_1+σ^1∂_2-m_3\boldsymbol1+m_4iσ^3\}Υ=\boldsymbol0,\\ \{-iσ^2∂_1+σ^1∂_2-m_3\boldsymbol1-m_4iσ^3\}\barΥ=\boldsymbol0.\tag{5B.100ab}$$ These equations are just the complex conjugate of one another, so only one of them is enough. It splits into $$\left\{\eqalign{ ∂_+α=\phantom-\tfrac12μγ,\quad ∂_+β=\phantom-\tfrac12μδ,\\ ∂_-γ=-\tfrac12\barμα,\quad ∂_-δ=-\tfrac12\barμβ. }\right.\tag{5B.101abcd}$$ The step numbers and the tangent vectors are $$E'_\pm=E_1\pm E_2=iσ_2⊗σ_\pm,\quad (E'^\pm=E^1\mp E^2=iσ^2⊗σ^\mp),\tag{5B.102}$$ $$\boldsymbol E'_\pm =σ_+⊗Υ^\timesσ_\pm\barΥ -σ_-⊗\barΥ^\timesσ_\pmΥ.\tag{5B.103}$$ From the induction formula (5B.81) and the orthonormality identities (5B.59) we get the components $$\hspace-3em\displaylines{x^1\pm x^2 =x^1_0\pm x^2_0-\tfrac12\langle\int_γ(iσ^2⊗σ^\mp) [(σ_+⊗Υ^\timesσ_-\barΥ -σ_-⊗\barΥ^\timesσ_-Υ)\dζ'^-+\\ +(σ_+⊗Υ^\timesσ_+\barΥ -σ_-⊗\barΥ^\timesσ_+Υ)\dζ'^+\rangle\\ =x^1_0\pm x^2_0+\int_γ (\tfrac18\bTr{σ^\mpΥ^\timesσ_-\barΥ +σ^\mp\barΥ^\timesσ_-Υ}\dζ'^-+\\ +\tfrac18\bTr{σ^\mpΥ^\timesσ_+\barΥ +σ^\mp\barΥ^\timesσ_+Υ}\dζ'^+),}\tag{5B.104a}$$ $$x^3 =x^3_0-\tfrac12\langle\int_γ(iσ^2⊗{\sf s}^3) [(σ_+⊗Υ^\timesσ_-\barΥ -σ_-⊗\barΥ^\timesσ_-Υ)\dζ'^-+\\ +(σ_+⊗Υ^\timesσ_+\barΥ -σ_-⊗\barΥ^\timesσ_+Υ)\dζ'^+\rangle\\ =x^3_0+\int_γ (\tfrac18\bTr{σ^3Υ^\timesσ_-\barΥ +σ^3\barΥ^\timesσ_-Υ)}\dζ'^-+\\ +\tfrac18\bTr{σ^3Υ^\timesσ_+\barΥ +σ^3\barΥ^\timesσ_+Υ}\dζ'^+),\tag{5B.104b}$$ $$\hspace-37pt\displaylines{x^4 =x^4_0-\tfrac12\langle\int_γ(σ^1⊗ i\boldsymbol1) [(σ_+⊗Υ^\timesσ_-\barΥ -σ_-⊗\barΥ^\timesσ_-Υ)\dζ'^-+\\ +(σ_+⊗Υ^\timesσ_+\barΥ -σ_-⊗\barΥ^\timesσ_+Υ)\dζ'^+\rangle\\ =x^4_0-i\int_γ (\tfrac18\bTr{Υ^\timesσ_-\barΥ -\barΥ^\timesσ_-Υ)}\dζ'^- +\tfrac18\bTr{Υ^\timesσ_+\barΥ -\barΥ^\timesσ_+Υ}\dζ'^+).}\tag{5B.104c}$$ With the help of the general expressions (5B.45) it gives the final formula [K96] $$x^1+x^2 =x^1_0+x^2_0+\int_γ (-β\barβ~\dζ'^-+δ\barδ~\dζ'^+),\tag{5B.105a}$$ $$x^1-x^2 =x^1_0-x^2_0+\int_γ (α\barα~\dζ'^--γ\barγ~\dζ'^+),\tag{5B.105b}$$ $$x^3+ix^4 =x^3_0+ix^4_0+\int_γ (-β\barα~\dζ'^-+δ\barγ~\dζ'^+),\tag{5B.105c}$$ $$x^3-ix^4 =x^3_0-ix^4_0+\int_γ (-α\barβ~\dζ'^-+γ\barδ~\dζ'^+).\tag{5B.105d}$$

Conclusion

The main result presented here is the demonstration of the way a Dirac-like equation can be used to construct a submanifold of a Euclidean space. It has been proven that as long as the solution of this equation belongs to the special Clifford group, differential $1$-forms, or equivalently vectors deduced from it by the rotation it represents, are found that give the immersion formula, and in a multitude of ways. This is true for arbitrary dimension, codimension and signature, while it was known only for a surface. The calculations though are almost intractable.

In the most difficult case, namely for a dimension $p>2$ and codimension $q>1,$ for a Euclidean normal space, the vectors don’t form an orthogonal frame, and are not conveniently rotations by the element of the Clifford group of a limited set of fundamental vectors, as usual in the moving frame method. This problem could be addressed with the extension of the Gauß map, for which explicit formulæ have been derived. Given any independent system of tangent vectors to the submanifold, the Gauß map could be computed, then by a lifting procedure that has been sketched, another solution of the Dirac equation could be deduced, that possess the desired nice properties. Then it would also be proved that any submanifold, not only conformal, admit a function in the special Clifford group that is a solution of a Dirac equation, and that gives an orthonormal set of tangent vector by the rotation of fixed vectors.

This lead us to the main application of this framework, the inverse scattering transform. It can be summed up as following : starting from an arbitrary Gauss map, we have the equation stating that it is indeed the Gauß map of something, in general it is a nonlinear partial differential equation of third order that can be simplified in special case and with additional data. The lift of the Gauß map is an independent system of vectors tangent to the submanifold, that can be rotated while keeping tangent. As the nonlinear equation is only about the Gauß map, it is not changed by these rotations. In the context of the inverse scattering transform, the equation is the considered partial nonlinear differential equation, the Weingarten equation about the tangent vectors is its representation by a linear system or Lax set, the rotation is a gauge transformation whose parameters are the spectral parameters. In the case of a surface in a $3$-dimensional Euclidean space, one gets many of the classical nonlinear equations in two dimensions in this way [S’03]. A much sought after generalisation is multidimensional equations. This work shows that for more than $2$ dimensions, there are as many spectral parameters as generators of the group of rotation, to each of them is associated a zero-curvature condition, compatibility condition of a Lax set whose cardinal equals the dimension. The calculative intricacies explain why it is a so challenging and discouraging problem.

It is true there is a well known multidimensional nonlinear partial differential equation, it is the self-dual, or anti self-dual, Yang Mills equation in Euclidean $4$ dimensions. But it is actually a degenerated case, analogous to the Liouville equation, that can be treated with only one spectral parameter, although multi parameters (twistors) were also introduced. The reason is, it is a conformally invariant equation, and we have seen that the immersion of a conformal manifold is much more easily described in the present framework.

All this has been studied locally, leaving the open problem whether that still holds globally, especially for the immersion of a closed submanifold, possibly non orientable.