General Spinor Representation of Submanifolds and Clifford Algebras
Framework

Claude Pierre Massé

Laboratoire de Physique Fondamentale, FRANCE

© Claude Pierre Massé 2026

5 November 2026

The Clifford Algebra language for describing the immersion of a (pseudo-)Riemannian manifold of dimension $p$ in $\Bbb R^n,$ $n\gt p$ through Cartan’s moving frame is developed and shown to be well suited. 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. For the sake of accessibility, the correspondance with usual differential geometry is spelled out. Then a Dirac like equation is derived, and it is investigated whether it is equivalent to the immersion condition like for a surface. Actually it is not and additional constraints are given. 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 to a full solution of the Dirac equation. Finally all the immersion formulæ known so far are derived, plus a new one for a $3$-manifold. The application of this framework to the multidimensional inverse scattering transform is discussed in the conclusion.

Latest version: https://phy.clmasse.com/spin-clifford-submanifold-framework.html

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. More recently the old Weierstraß-Enneper induction formula has been refurbished and generalised by using the 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. This is due to formidable algebraic barriers and steeply increasing difficulties that need to be tackled with more powerful calculational tools. Clifford algebras provide such tools, as demonstrated here, when used together with Cartan’s moving frame.

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 $$\pmatrix{φ& -\barχ\\χ& \phantom-\barφ},\quadφ,χ∈\Bbb C.$$ 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 $$\varPsi=\pmatrix{φ'+iφ''& -χ'+iχ''\\χ'+iχ''& \phantom-φ'-iφ''} =φ'E+χ''E_1-χ'E_2-φ''E_{12}=ψ^IE_I∈{\rm C}ℓ_{2,0},\quadψ^I∈\Bbb R.$$ It is but a compacted version, that we expand by using the isomorphism $${\rm C}ℓ_{n-1,0}↪{\rm C}ℓ_{n,0}^0,$$ generated by $$E_i↦E_{ni},$$ where ${\rm C}ℓ^0$ is the even subalgebra of ${\rm C}ℓ.$ Then $\varPsi$ 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. Introducing bigger Clifford algebras and the associated spinor, we expect to be able to treat higher dimensional cases. 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.

We shall 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. Since it isn’t the more usual language of differential geometry, this section is appended by a lexicon to make the necessary translations. The central part is the derivation of the Dirac equation starting from the data of the submanifold immersed in a flat space, and how it reduces is special cases, including how it gives the known equation for a surface. §3. A essential concept in these problems is the Gauß map, that is worked out and extended in §4. For it we indicate how it can be lifted to a full solution of the Dirac equation and obtain the constaint it must satisfy to the be Gauß map of a submanifold. The last section is devoted to recover the concrete formula for particular cases that are found in the literature, with the addition of a new one for a higher dimension submanifold §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 here 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 ${\rm C}ℓ_{n_+,n_-},$ 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 ${\rm C}ℓ_{n_+,n_-}$ 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 ${\rm C}ℓ_n,$ 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 or imaginaries $E_1,⋯,E_n,$ 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 : $$E_iE_j=-E_jE_i,\quad E_iE_i=±E.\tag{1.1}$$ 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 $$C=cE+c^iE_i+c^{ij}E_{ij}+⋯\tag{1.2}$$ By convention, when there are several indices, they are all distinct.

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 $E_iE_j⋯E_m$ is $E_m⋯E_jE_i.$ 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 : $$η_i=-E_iE_i,\quad E^i=η_iE_i,\tag{1.6}$$ where the indicator $η_i$ 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 $$E_I=E_{i\ph\phℓ}^{\ph jk},\quad E^I=E^{ℓ\ph\ph i}_{\ph kj} =(-)^{|I|(|I|-1)/2}η_iη_jη_kη_ℓ E_I=:η_IE_I,\tag{1.8}$$ so that in the same way $$E_IE^I=E^IE_I=(-)^{|I|}E.\tag{1.9}$$ Beware the conventions for $η$ are different from usual for the Minkowski metric, in particular there is never summation in its index.

We use some shorthand notations : $$(c^KE_K)^I=c^I,\quad⟨C⟩_r=\sum_{|I|=r}(C)^IE_I\tag{1.10}$$ are the $I$ component and the $r$ grade of a Clifford number. $$E_P=E_{1⋯p},\quad E_N=E_{1⋯n}\tag{1.11}$$ are the unit pseudo scalars of ${\rm C}ℓ_p$ and ${\rm C}ℓ_n,$ while $$E_Q=E_{p+1⋯n}.\tag{1.12}$$

The $0$ grade $$⟨C⟩:=⟨C⟩_0\tag{1.13}$$ has the properties of the trace. It is linear, and $$⟨AB⟩ =⟨(a^IE_I)(b^JE_J)⟩ =a^Ib^J⟨E_IE_J⟩ =b^Ja^I⟨E_JE_I⟩ =⟨BA⟩,\tag{1.14}$$ $$⟨ABC⟩=⟨CAB⟩=⟨BCA⟩,\tag{1.15}$$ while for all $r,$ $$⟨\tilde C⟩_r=(-)^{r(r-1)/2}⟨C⟩_r.\tag{1.16}$$

The most important structure of the Clifford algebras for us, and that generalises the spinor $\varPhi,$ 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 $$\boldsymbol V'=\varPsi^{-1}V\varPsi.\tag{1.17}$$ The inverse comes first because we reserve the symbol $\varPsi$ for the wave function. But that is only a matter of convention, the mathematics are the same by swapping $\varPsi^{-1}\leftrightarrow\varPsi$ 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 : $$\varPsi(V)=\tilde\varPsi V\varPsi.\tag{1.18}$$ Since $\tilde\varPsi\varPsi$ is a multiple of the unity, this introduces an additional scaling. Finally the subgroup defined by $\tilde\varPsi\varPsi=E$ 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 subalgebra 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 $1,2,⋯,p,$ Latin ones $a,b,⋯, r,s,⋯$ for $p+1,⋯,n,$ while middle Latin ones $i,j,k,ℓ$ are used for the full range $1,2,⋯,n.$ 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 $ξ^μ,~μ=1,⋯,p.$ ${\cal D}$ is immersed in $\Bbb R^n$ by the function $$\boldsymbol x:{\cal D}\to\Bbb R^n,\quad ξ=(ξ^1,⋯,ξ^p)↦\boldsymbol x(ξ)=(x^1,⋯,x^n).\tag{2A.1}$$ To each point of the immersed space, that is for each value of $(ξ^1,⋯,ξ^p),$ is attached an adapted orthogonal frame $(\boldsymbol e_1,⋯,\boldsymbol e_n)^t$ 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 $$\boldsymbol e_i⋅\boldsymbol e^j=ρ^2δ_i^j.\tag{2A.2}$$ The operation $⋅$ is the associated bilinear form of $\Bbb R^n$ and $ρ=\e^u$ is a real function over ${\cal D}.$ We call it a conformal frame. 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]. Differential $1$-forms $ω$ over ${\cal D}$ are defined through the structure equations of the space ($\d=\dξ^μ∂_μ$) $$\left\{\eqalign{ \d\boldsymbol x& =ω^i\boldsymbol e_i,\\ \d\boldsymbol e_i& =ω_i^k\boldsymbol e_k, }\right.\tag{2A.3ab}$$ which, multiplied by $\boldsymbol e^j$ on the right, yields $$\left\{\eqalign{ ω^j& =ρ^{-2}\d\boldsymbol x⋅\boldsymbol e^j,\\ ω_i^j& =ρ^{-2}\d\boldsymbol e_i⋅\boldsymbol e^j. }\right.\tag{2A.4ab}$$ Here we have $$ω^a=0.\tag{2A.5}$$ Differentiating the orthonormality conditions (2A.2) we get $$ω_i^j+η_{ij}ω_j^i=2\d u~δ_i^j\quad\text{(no sum.)}\tag{2A.6}$$

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 : $$\left\{\eqalign{ \dω^i& =ω^α∧ω^i_α\\ \dω^i_j& =ω^k_j∧ω^i_k. }\right.\tag{2A.7ab}$$

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 $$\left\{\eqalign{ \dω^α& =ω^γ∧ω^α_γ\\ \dω^α_β& =ω^γ_β∧ω^α_γ+θ^α_β, }\right.\tag{2A.8ab}$$ in which $θ^α_β=ω^a_β∧ω^α_a$ is the curvature $2$-form. Through differentiation of these equations, substituting the differentials $\dω$ by their expression in the same equations, we get $$\left\{\eqalign{ ω^γ∧θ_γ^α& =0\\ \dθ^α_β -ω^γ_β∧θ^α_γ+θ^γ_β∧ω^α_γ& =0 =:{\frak d}θ^α_β. }\right.\tag{2A.9ab}$$ These are the Bianchi identities. Similar identities are obtained in the normal space by defining a second $2$-form $τ$ : $$\dω^a_b=ω^c_b∧ω^a_c+τ^a_b,\tag{2A.10}$$ that is $$τ^a_b=ω^γ_b∧ω^a_γ,\tag{2A.11}$$ $$\dτ^a_b-ω^c_b∧τ^a_c+τ^c_b∧ω^a_c=0 =:{\frak d}τ^a_b.\tag{2A.12}$$ The $2$-form $θ$ actually corresponds to the Riemann tensor, and $τ$ is called the Riemannian or Gaußian torsion. (Here they are the same because the ambient space is flat.) The symmetries of these forms are $$θ^α_β+η_{αβ}θ^β_α,\tag{2A.13}$$ $$τ^a_b+η_{ab}τ^b_a,\tag{2A.14}$$

The fundamental forms of the submanifold are given as usual by the differential forms $${\rm I}=\d\boldsymbol x⋅\d\boldsymbol x =ρ^2ω^αω_α,\tag{2A.I}$$ $${\rm II}_a=\d(ρ^{-1}\boldsymbol e_a)⋅\d\boldsymbol x =ρ\,ω^α_aω_α,\tag{2A.II}$$ $${\rm III}_{ab}=\d(ρ^{-1}\boldsymbol e_a)⋅\d(ρ^{-1}\boldsymbol e_b) =ω^α_aω_{bα}.\tag{2A.III}$$

2B. Clifford formulation

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

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 $$Z:=ρ^{-1}\d\varPsi\tilde\varPsi=\d\varPsi\varPsi^{-1}.\tag{2B.4}$$ To work out this expression, let us write $$\varPsi=ρ^{1/2}\hat\varPsi,\tag{2B.5}$$ where $\hat\varPsi∈ {\rm Spin}(n).$ Inserting it in the expression $Z,$ we get $$ρ^{-1}\d\varPsi\tilde\varPsi=\tfrac12\d u+\d\hat\varPsi\hat\varPsi^{-1}.\tag{2B.6}$$ Now $\d\hat\varPsi\hat\varPsi^{-1}∈{\frak spin}(n)=\span(E_i^{\ph j}/2)$ is a bivector. We denote it by $\hat Z.$ Finally $$Z=ρ^{-1}\d\varPsi\tilde\varPsi=\tfrac12\d u+\hat Z=:\tfrac12\d u+\tfrac12z_i^jE_j^{\ph i}.\tag{2B.7}$$

Now the second structure equation of the group of motions (2B.3b) reads $$\d(\tilde\varPsi E_i\varPsi)=\d\tilde\varPsi E_i\varPsi+\tilde\varPsi E_i\d\varPsi=ω_i^k\tilde\varPsi E_k\varPsi.\tag{2B.8}$$ Multiplying on the left by $ρ^{-1}\varPsi$ and on the right by $ρ^{-1}\tilde\varPsi$ it becomes $$(ρ^{-1}\d\varPsi\tilde\varPsi)^\Tilde E_i+E_i(ρ^{-1}\d\varPsi\tilde\varPsi) =\tilde ZE_i+E_iZ =ω_i^kE_k.\tag{2B.9}$$ As $\hat Z$ is a bivector the last equation is equivalent to $$\d uE_i-[\hat Z,E_i]=ω_i^kE_k.\tag{2B.10}$$ 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 $$\eqalign{ 2z_i^{j≠i}& =ω_i^j\\ 2Z& =\d u+\tfrac12ω_i^jE_j^{\ph i}. }\tag{2B.11ab}$$ The second structure equation of the group of motions (2B.3b) is therefore $$\d\boldsymbol E_i=\d u\boldsymbol E_i+2z_i^{k≠i}\boldsymbol E_k.\tag{2B.12}$$ From here on, we no longer explicitely write conditions like $k≠i,$ they are implied and $z^i_i$ is not defined. In pure spinor notation, this equation is equivalent to the obvious relation $$\d\varPsi=Z\varPsi,\tag{2B.GW}$$ also known as the Gauß-Weingarten equation. From the compatibility condition of the system $$\left\{\eqalign{ ∂_μ\varPsi=Z_μ\varPsi,\\ ∂_ν\varPsi=Z_ν\varPsi, }\right.\tag{2B.13ab}$$ as usual by cross differentiating, the scalar terms cancel and we get the so-called zero-curvature conditions $$∂_μ\hat Z_ν-∂_ν\hat Z_μ =[\hat Z_μ,\hat Z_ν],\tag{2B.14}$$ or in components, $$∂_μ z_{i/ν}^j-∂_ν z_{i/μ}^j =2z_{i/μ}^kz_{k/ν}^j-2z_{i/ν}^kz_{k/μ}^j.\tag{2B.15}$$ 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 $$Z+\tilde Z=\d u,\tag{2B.16}$$ then $$z_i^j+η_{ij}z_j^i=0,\quad i≠j,\tag{2B.17}$$ 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 $$∂_μ\hat Z^+_ν-∂_ν\hat Z^+_μ =[\hat Z^+_μ,\hat Z^+_ν]+[\hat Z^-_μ,\hat Z^-_ν],\tag{2B.GR}$$ $$∂_μ\hat Z^-_ν-[\hat Z^+_μ,\hat Z^-_ν] =∂_ν\hat Z^-_μ-[\hat Z^+_ν,\hat Z^-_μ].\tag{2B.CM}$$ 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 $\varPsi(ξ),$ 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 $$R_{μν}+F_{μν} :=-∂_μ\hat Z^+_ν+∂_ν\hat Z^+_μ+[\hat Z^+_μ,\hat Z^+_ν] =-[\hat Z^-_μ,\hat Z^-_ν],\tag{2B.23}$$ where $R_{μν}$ made of all the terms in $E^{\phβ}_α$ corresponds to $θ^α_β,$ and $F_{μν}$ made of all the terms in $E^{\ph b}_a$ corresponds to $τ^a_b.$ We have the usual expression $$R_{μν}+F_{μν} =[∂_μ-\hat Z^+_μ,∂_ν-\hat Z^+_ν],\tag{2B.24}$$ so that the Jacobi identity $$[∂_λ-\hat Z^+_λ,R_{μν}+F_{μν}] +[∂_μ-\hat Z^+_μ,R_{νλ}+F_{νλ}] +[∂_ν-\hat Z^+_ν,R_{λμ}+F_{λμ}]=\boldsymbol0\tag{2B.25}$$ directly gives the second Bianchi identity $$\phantom=∂_λ(R_{μν}+F_{μν}) +∂_μ(R_{νλ}+F_{νλ}) +∂_ν(R_{λμ}+F_{λμ})\\ =[\hat Z^+_λ,R_{μν}+F_{μν}] +[\hat Z^+_μ,R_{νλ}+F_{νλ}] +[\hat Z^+_ν,R_{λμ}+F_{λμ}],\tag{2B.B2}$$ which is in two independent parts $$∂_λR_{μν}-[\hat Z^∥_λ,R_{μν}] +∂_μR_{νλ}-[\hat Z^∥_μ,R_{νλ}] +∂_νR_{λμ}-[\hat Z^∥_ν,R_{λμ}]=\boldsymbol0,\tag{2B.B2r}$$ $$∂_λF_{μν}-[\hat Z^⊥_λ,F_{μν}] +∂_μF_{νλ}-[\hat Z^⊥_μ,F_{νλ}] +∂_νF_{λμ}-[\hat Z^⊥_ν,F_{λμ}]=\boldsymbol0,\tag{2B.B2f}$$ if $\hat Z^∥$ is made of the terms in $E^{\phβ}_α$ and $\hat Z^⊥$ of the terms in $E^{\ph b}_a.$

Finally, the fundamental forms become $${\rm I}=-⟨\d\boldsymbol X\d\boldsymbol X⟩ =ρ^2ω^αω_α,\tag{2B.I}$$ $${\rm II}_a=-⟨\d(ρ^{-1}\boldsymbol E_a)\d\boldsymbol X⟩ =2ρz^α_{a/μ}\dξ^μω_α,\tag{2B.II}$$ $${\rm III}^b_a=-⟨\d(ρ^{-1}\boldsymbol E_a)\d(ρ^{-1}\boldsymbol E^b)⟩ =4z^α_{a/μ}z^b_{α/ν}\dξ^μ\dξ^ν.\tag{2B.III}$$

2C. Differential geometry lexicon

There are different languages in differential geometry, the one of Cartan being not the more usual. In order that the following be accessible for most reader, we review the different concepts and objects, adapted to conformal frames, and the relations between them.

On a manifold, we use a coordinate system to label every points. For all purposes here, the coordinates are identical to $ξ^μ$ for the submanifold. The $μ$ type indices are then called coordinate indices (or holonomic indices.) In addition, at each point is attached an adapted conformal frame $\boldsymbol E_i,$ that is, the $\boldsymbol E_α$ are tangent to the submanifold, then the $\boldsymbol E_a$ are normal to it. The $α$ type indices are called Lorentz indices (or anholonomic indices) by analogy with space-time in general relativity. While the coordinate indices are lowered and raised with the metric tensor in the usual way, the Lorentz indices are lowered and raised like in (1.6). With the differential forms $ω^α=ω^α_{/μ}\dξ^μ$ we define another system of tangent vectors, so called coordinate vectors : $$\boldsymbol E_{/μ}=ω^α_{/μ}\boldsymbol E_α.\tag{2C.1}$$ In this context, the differential forms are called the soldier forms often denoted by $θ^α,$ the coframe, or more recently the fiducial tensor denoted $\underline h.$ We easily verify that $$\boldsymbol E_{/μ}\boldsymbol E_{/ν} +\boldsymbol E_{/ν}\boldsymbol E_{/μ} =2ρ^2ω^α_{/μ}ω_{α/ν} =2ρ^2ω_{α/μ}ω^α_{/ν} =2g_{μν}\tag{2C.2}$$ where $g_{μν}$ is the induced (Riemannian) metric of the submanifold, or fundamental tensor. Now we consider $ω^{/μ}_α$ that is the inverse matrix of $ω^α_{/μ},$ Thus $$ω^{/μ}_βω^α_{/μ}=δ^α_β,\quad ω^α_{/ν}ω^{/μ}_α=δ^μ_ν.\tag{2C.3}$$ Clearly $$g_{μ\nu}ω^{α/μ}ω^{/ν}_β=ρ^2δ^α_β,\tag{2C.4}$$ so that $ω^{/μ}_α,$ denoted $λ^{\phantom{α|}μ}_{α|}$ and called orthogonal ennuple, is the components $μ$ of the vector $α$ of a conformal frame. It is also called the Vielbein or many legs (sometimes tetrad in $4$ dimensions) denoted $h^μ_α,$ or the reciprocate coframe.

As for the other set of differential forms $ω^α_β=ω^α_{β/μ}\dξ^μ,$ they are sometimes called the spin connection or more recently the fiducial connection, while the combination $ω^α_{β/μ}E_α E^β$ receive the name of Fock-Ivanenko coefficients denoted $\varGamma_μ,$ or spinor affine connection.

The absolute or covariant derivative, in the case of an immersion, is the partial derivative with respect to the coordinates of a vector, then projected upon the tangent space. Since we have $$∂_μ\boldsymbol E_α=ω^i_{α/μ}\boldsymbol E_i,\tag{2C.5}$$ it is given by $$∇_μ\boldsymbol E_α=ω^γ_{α/μ}\boldsymbol E_γ =∂_μ\boldsymbol E_α-ω^a_{α/μ}\boldsymbol E_a.\tag{2C.6}$$ The so called spinorial covariant derivative is defined by $$D_μψ=(∂_μ-\tfrac14ω^α_{β/μ}E_α E^β)ψ.\tag{2C.7}$$ Of course in our case, substituting the function $\varPsi$ for $ψ,$ it vanishes. With the Lorentz indices we have $$∇_β\boldsymbol E_α :=ω^{/μ}_β∇_μ\boldsymbol E_α =ω^{/μ}_βω^γ_{α/μ}\boldsymbol E_γ =:γ^γ_{αβ}\boldsymbol E_γ.\tag{2C.8}$$ The $$γ^γ_{αβ}=ω^{/μ}_βω^γ_{α/μ}\tag{2C.9}$$ are called the Ricci rotation coefficients, like the spin connection they are antisymmetric with respect to the leftmost indices, and are generally invariant, having no coordinate indices. On the other hand with the coordinate indices we have $$\eqalign{∇_μ\boldsymbol E_{/ν} & =∇_μ(ω^α_{/ν}\boldsymbol E_α) =(∂_μω^α_{/ν} +ω^β_{/ν}ω^γ_{β/μ})\boldsymbol E_α\\ & =(∂_μω^α_{/ν} +ω^β_{/ν}ω^γ_{β/μ})ω^{/λ}_α\boldsymbol E_{/λ} =:\varGamma^λ_{\phantom\phμν}\boldsymbol E_{/λ}.}\tag{2C.10}$$ The $\varGamma^λ_{\phantom\phμν}$ are called the Christoffel ($3$-index) symbols of the second kind when they are denoted ${μν\braceλ}$ or now ${λ\braceμν},$ the connection coefficients, or still the Levi-Civita connection. Their relation with the spin connection is then $$\varGamma^λ_{\phantom\phμν} =(∂_μω^α_{/ν} +ω^β_{/ν}ω^γ_{β/μ})ω^{/λ}_α,\tag{2C.11}$$ and they are symmetric with respect to $μ,ν$ from the structure equation of the group of motion. They can also be expressed in the classical way $$\varGamma^λ_{\phantom\phμν}=\tfrac12 g^{λκ}( ∂_μ g_{κν}+∂_ν g_{κμ}-∂_κ g_{μν}).\tag{2C.12}$$ The verification is straightforward but lengthy and not transparent. Inversely we have $$γ^γ_{αβ} =-ω^{/μ}_αω^{/ν}_β(∂_νω^γ_μ -\varGamma^λ_{\phantom\phμν}ω^γ_{/λ}).\tag{2C.13}$$

We proceed to the next order, with the commutators : $$[∇_μ,∇_ν]\boldsymbol E_α =(∂_μω^β_{α/ν} -∂_νω^β_{α/μ} +ω^γ_{α/ν}ω^β_{γ/μ} -ω^γ_{α/μ}ω^β_{γ/ν}) \boldsymbol E_β =:r^β_{α/μν}\boldsymbol E_β,\tag{2C.14}$$ the $r^β_{α/μν}$ is the spin curvature given by $$r^β_{α/μν}=∂_μω^β_{α/ν} -∂_νω^β_{α/μ} +ω^γ_{α/ν}ω^β_{γ/μ} -ω^γ_{α/μ}ω^β_{γ/ν}.\tag{2C.15}$$ It is manifestly antisymmetric with respect to its Lorentz indices as well as with respect its the coordinate indices. If instead we calculate $$[∇_μ,∇_ν]\boldsymbol E_{/λ} =:R^κ_{\phantom\phλμν}\boldsymbol E_{/κ}\tag{2C.16}$$ we get $$\eqalign{ R^κ_{\phantom\phλμν} & =ω^{/κ}_α[∂_μ(ω^β_{/λ}ω^α_{β/ν})-∂_ν(ω^β_{/λ}ω^α_{β/μ})+\\ & \phantom=+(∂_νω^β_{/λ}+ω^γ_{/λ}ω^β_{γ/ν})ω^α_{β/μ} -(∂_μω^β_{/λ}+ω^γ_{/λ}ω^β_{γ/μ})ω^α_{β/ν}]\\ & =ω^α_{/λ}ω^{/κ}_β(∂_μω^β_{α/ν}-∂_νω^β_{α/μ} +ω^γ_{α/ν}ω^β_{γ/μ}-ω^γ_{α/μ}ω^β_{γ/ν})\\ & =ω^α_{/λ}ω^{/κ}_βr^β_{α/μν}.}\tag{2C.17}$$ This is called the Riemann symbols of the second kind, the (mixed) Riemann tensor, or the Riemannian curvature. Its relation with the spin curvature is simpler because it is a tensor, while the connection is not. From its contractions we have other tensors : $$R_{μν}=δ^λ_κ R^κ_{\phantom\phμλν} =ω^α_{/μ}ω^{/λ}_βr^β_{α/λν}\tag{2C.18}$$ is called the Ricci tensor, and $$R=g^{μν}R_{μν} =r^β_{α/μν}ω^{α/ν}ω^{/μ}_β\tag{2C.19}$$ is called the scalar curvature. We then have the Einstein tensor $$G_{μν}:=R_{μν}-\tfrac12Rg_{μν} =ω^α_{/μ}ω^{/λ}_β(r^β_{α/λν} -\tfrac12ω^{γ/κ}ω_{α/ν}r^β_{γ/λκ}).\tag{2C.20}$$ For $p=3$ and $p=4$ $$\{r^α_{β/μν} E^{\phβ}_α,E^{\ph/μν}_{/σ}\}=8G_{στ}E^{/τ}.\tag{2C.21}$$

To be complete, we mention the Lie bracket $$\eqalign{[\boldsymbol E_α,\boldsymbol E_β] & :=ω^{/μ}_α∇_μ\boldsymbol E_β -ω^{/μ}_β∇_μ\boldsymbol E_α\\ & =(γ^γ_{βα}-γ^γ_{αβ})\boldsymbol E_γ =(ω^{/ν}_α∂_νω^{/μ}_β -ω^{/ν}_β∂_νω^{/μ}_α) ω^γ_{/μ}\boldsymbol E_γ,}\tag{2C.22}$$ where $$∂_α:=ω^{/μ}_α∂_μ\tag{2C.23}$$ is called the intrinsic derivative.

§3. Dirac equation

Now we are ready to 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. Derivation

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 $ω^α$ are constant multiples of $\dξ^α.$ This is not true in higher dimensions. We adopt the reverse way and start from these forms considered arbitrary, since the other forms can be deduced from them under some condition [E §55] that are introduced when needed.

From the first structure equations of the group of motions (2A.7a) written as $$\left[\eqalign{ ∂_μω^γ_{/ν}-∂_νω^γ_{/μ} & =ω^α_{/μ}ω^γ_{α/ν} -ω^α_{/ν}ω^γ_{α/μ},\\ 0& =ω^γ_{/μ}ω^a_{γ/ν} -ω^γ_{/ν}ω^a_{γ/μ}, }\right.\tag{3A.1}$$ and with (2C.9) we deduce easily $$\left[\eqalign{ γ^γ_{αβ}-γ^γ_{βα} & =ω^{/μ}_αω^{/ν}_β (∂_μω^γ_{/ν}-∂_νω^γ_{/μ}),\\ γ^a_{αβ}-γ^a_{βα}& =0, }\right.\tag{3A.2ab}$$ while from (2A.6) we have $$γ^γ_{γα}=ω^{/μ}_α∂_μ u \quad(\text{not summed on $γ.$})\tag{3A.3}$$ We assume that the derivatives of $~\varPsi$ correspond to the forms of the submanifold, that is $$z^i_j=\tfrac12ω^i_j.\tag{3A.4}$$ To exploit the symmetry, we compute the contraction $$\eqalign{ ω^{/μ}_α∂_μ\varPsi & =ω^{/μ}_α\left(\tfrac12∂_μ u +\tfrac12 z^a_{b/μ}E^{\ph b}_a +\tfrac12 z^γ_{β/μ}E^{\phβ}_γ +z^γ_{a/μ}E^{\ph a}_γ\right)\varPsi\\ & =\left(\tfrac12ω^{/μ}_α[∂_μ u +z^a_{b/μ}E^{\ph b}_a] +\tfrac14γ^γ_{βα}E^{\phβ}_γ +\tfrac12γ^γ_{aα}E^{\ph a}_γ\right)\varPsi.}\tag{3A.5}$$ Now we construct the Dirac operator $$ω^{/μ}_α E^α∂_μ\varPsi =\left(\tfrac12ω^{/μ}_α E^α[∂_μ u +z^a_{b/μ}E^{\ph b}_a] +\tfrac14γ^γ_{βα}E^α E^{\phβ}_γ +\tfrac12γ^γ_{aα}E^α E^{\ph a}_γ\right)\varPsi,\tag{3A.6}$$ and get the equation $$\Big\{ω^{/μ}_α E^α[∂_μ-\tfrac12∂_μ u-A_μ] +\tfrac12\sum_{γ≠α}γ^γ_{αγ}E^α -\tfrac18(γ^γ_{αβ}-γ^γ_{βα})E^{\phαβ}_γ +M\Big\}\varPsi=\boldsymbol0\tag{3A.7}$$ where we have used the symmetry of $γ^γ_{aα}$ (3A.2b) in the last term, and we have set $$\displaylines{m_a:=\sum_γγ^γ_{aγ },\quad M:=m_aE^a,\\ A_μ:=\tfrac12z^a_{b/μ}E^{\ph b}_a.}\tag{3A.8abc}$$ Now if the determinant $$|ω^α_{/μ}|=:\sqrt h,\tag{3A.9}$$ we have $$∂_μω^α_{/ν}ω^{/ν}_α =∂_μ\sqrt h/\sqrt h\tag{3A.10}$$ so that calculating from (3A.2a) and (3A.3) $$\sum_{γ≠α}γ^γ_{αγ} =(p-1)ω^{/μ}_α∂_μ u +∂_μ(ω^{/μ}_α\sqrt h)/\sqrt h.\tag{3A.11}$$ Remark that $$g:=|ρ^2ω^α_{/μ}ω_{α/ν}|=\e^{2pu}h.\tag{3A.12}$$ With all these together we get the Dirac equation $$\begin{multline}\hspace6em \Big\{ω^{/μ}_α E^α[∂_μ+\tfrac12(p-2)∂_μ u-A_μ] +\tfrac12∂_μ(ω^{/μ}_α\sqrt h)/\sqrt hE^α+\\ -\tfrac18(γ^γ_{αβ}-γ^γ_{βα})E^{\phαβ}_γ +M\Big\}\varPsi=\boldsymbol0. \hspace6em\tag{3A.13}\end{multline}$$

This equation is still not quite satisfactory because of the term in $∂_μ u$ that vanishes for a surface only, making it nonlinear since $u$ gives the norm of $\varPsi.$ It is the grain of sand that must be removed to make the machinery work in multidimensions. For this purpose we introduce another set of parameters : $$ω^{/μ}_α=:\e^{-au}ω'^{/μ}_α,\quad ω^α_{/μ}=:\e^{au}ω'^α_{/μ},\tag{3A.14}$$ $$\sqrt h=\e^{pau}\sqrt h',\tag{3A.15}$$ $$\displaylines{γ'^γ_{αβ}-γ'^γ_{βα} :=ω'^{/μ}_αω'^{/ν}_β(∂_μω'^γ_{/ν}-∂_νω'^γ_{/μ}),\\ γ^γ_{αβ}-γ^γ_{βα} =\e^{-au}(γ'^γ_{αβ}-γ'^γ_{βα}),\quad α,β,γ\text{ distinct,}}\tag{3A.16}$$ where $a$ is a real constant. Then $$\begin{multline}\hspace4em \e^{-au}\Big\{ω'^{/μ}_αE^α[∂_μ+\tfrac12(p-2)∂_μu-A_μ] +\tfrac12ω'^{/μ}_α(p-1)a∂_μ u E^α+\\ +\tfrac12∂_μ(ω'^{/μ}_α\sqrt h')/\sqrt h'E^α -\tfrac18(γ'^γ_{αβ}-γ'^γ_{βα})E^{\phαβ}_γ +M'\Big\}\varPsi=\boldsymbol0 \hspace4em\tag{3A.17}\end{multline}$$ where now $$M'=\e^{au}M,\tag{3A.18}$$ and the nonlinear term disappears if $a$ satisfies $$(p-1)a+p-2=0.\tag{3A.19}$$ The final equation is $$\hspace-2em\Big\{ω'^{/μ}_α E^α[∂_μ-A_μ] +\tfrac12∂_μ(ω'^{/μ}_α\sqrt h')/\sqrt h'E^α -\tfrac18(γ'^γ_{αβ}-γ'^γ_{βα})E^{\phαβ}_γ +M'\Big\}\varPsi=\boldsymbol0,\tag{3A.20}$$ or alternatively $$\begin{multline}\hspace6em \Big\{ω'^{/μ}_α E^α[∂_μ +\tfrac12ω'^{/ν}_β(∂_μω'^β_{/ν}-∂_νω'^β_{/μ})+\\ +\tfrac18\sum_{β,γ≠α} ω'^{/ν}_β(∂_μω'^γ_{/ν}-∂_νω'^γ_{/μ})E^{\phβ}_γ -A_μ]+M'\Big\}\varPsi=\boldsymbol0 \hspace6em\tag{3A.21}\end{multline}$$ using (2C.3), (3A.2a), and (3A.10). It is the same as the Dirac equation in a Riemannian space [F29, W29a, W29b], and also the one obtained by the restriction of a parallel spinor [F98, BLR’17]. This shows the geometrical origin of the physical Dirac equation.

When $\varPsi$ represents a submanifold, it is a solution of this equation. But does all solutions of this equation represent a submanifold? To analyze it more easily, we evaluate the derivative and break it down in the parallel $∥,$ normal $⊥,$ and odd $−$ parts, that should all be satisfied. To this end, similarly to $γ'$ we introduce $$c'^γ_{αβ} =ω'^{/μ}_βz^γ_{α/μ}\tag{3A.22}$$ that we use to stand for the components of the logarithmic derivatives of $\varPsi.$ The result is $$\hspace-3em\tfrac12ω'^{/μ}_α E^α∂_μ u +\tfrac12c'^γ_{βα}E^α E^{\phβ}_γ +\tfrac12∂_μ(ω'^{/μ}_α\sqrt h')/\sqrt h'E^α -\tfrac18(γ'^γ_{αβ}-γ'^γ_{βα})E^{\phαβ}_γ =\boldsymbol0.\tag{3A.23∥}$$ $$ω'^{/μ}_α E^α[\tfrac12z^a_{b/μ}E^{\ph b}_a-A_μ] =\boldsymbol0.\tag{3A.23⊥}$$ $$c'^β_{aα}E^α E^{\ph a}_β+M'=\boldsymbol0.\tag{3A.23−}$$ In (3A.23⊥) $A_μ$ is given by $$A_μ=a^a_{b/μ}E^{\ph b}_a.\tag{3A.24}$$ We see immediately that if $~\varPsi$ is a solution then, breaking further down in vector and trivector $$\sum_{γ≠α}c'^γ_{αγ} =\tfrac12(ω'^{/μ}_α∂_μ u +∂_μ(ω'^{/μ}_α\sqrt h')/\sqrt h') =\frac12\sum_{γ≠α}γ'^γ_{αγ}\tag{3A.25∥1}$$ $$c'^γ_{βα}+c'^α_{γβ}+c'^β_{αγ} =\tfrac12(γ'^γ_{βα}+γ'^α_{γβ}+γ'^β_{αγ})\quad α,β,γ\text{ distinct}.\tag{3A.25∥3}$$ $$z^a_{b/μ}=a^a_{b/μ}\tag{3A.25⊥}$$ $$\sum_α c'^α_{aα}E^a=M'\tag{3A.25−1}$$ $$c'^a_{αβ}=c'^a_{βα}\quadα≠β.\tag{3A.25−3}$$ While the data $ω'^α_{/μ}$ give unambiguously the $γ',$ the Dirac equation only fixes some sums of the $c'.$ For a surface there is only one term in the sum (3A.25∥1), and there is no $γ'$ with three distinct indices, so that the Dirac equation is logically equivalent to the immersion condition. This is no longer the case for $p>2,$ even though this has been wrongly claimed in [LR’10] for $p=3.$ The function $\varPsi$ must fullfill additional constaints.

In addition to the general case, we consider three special cases that are simpler.

• Case 0 :
  For a surface, $p=2,$ $a=0$ and there is no trivector term faute de combattants. There always exists a conformal coordinate system in which the metric is scalar, that is $$g_{μν}=\rho^2ω^α_{/μ}ω_{α/ν} =\rho^2δ_{μν},\tag{3A.26}$$ $$ω^α_{/μ}=ω'^α_{/μ}=δ^α_μ,\tag{3A.27}$$ leaving $$\big\{E^μ[∂_μ-A_μ]+M\big\}\varPsi=\boldsymbol0.\tag{3A.28}$$ This equation is the one that have been extensively studied, see [I] and references therein.


• Case i :
  For a conformally flat submanifold of any dimension and codimension, if a conformal coordinate system is chosen, the intrinsic metric is scalar and we then have $$ω^α_{/μ}=\e^vδ^α_μ,\quad ω^{/μ}_α=\e^{-v}δ^μ_α,\tag{3A.29}$$ which gives $$γ^γ_{αβ}-γ^γ_{βα}=0 \quadα,β,γ\text{ distinct,}\tag{3A.30}$$ $$\sqrt h=\e^{pv}.\tag{3A.31}$$ We put these expressions into equation (3A.13) and get $$\Big\{\e^{-v}E^μ[∂_μ+\tfrac12(p-2)∂_μ u +\tfrac12(p-1)∂_μ v-A_μ] +M\Big\}\varPsi=\boldsymbol0.\tag{3A.32}$$ Owing to (3A.19), we should have $v=au$ to eliminate the term in $∂_μ u.$ We then get $$ω^α_{/μ}=\e^{au}δ^α_μ,\quad ω^{/μ}_α=\e^{-au}δ_α^μ,\tag{3A.33}$$ $$ω'^α_{/μ}=δ^α_μ,\quad ω'^{/μ}_α=δ_α^μ,\tag{3A.34}$$ and the equation $$\Big\{E^μ[∂_μ-A_μ]+M'\Big\}\varPsi=\boldsymbol0.\tag{3A.35}$$ Furthermore, from (3A.30) we deduce $$γ^γ_{αβ}=0,\quadα,β,γ\text{ distinct}\tag{3A.36}$$ by making use of the relation (2A.6) $γ^γ_{αβ}=-η_{αβ}γ^α_{γβ}.$ 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. From (3A.2a) and (3A.3) we moreover have $$γ^α_{βα}=\e^{-au}∂_β u/(p-1),\tag{3A.37}$$ which does not depend on $α.$ Hence the additional constraints are $$\eqalign{z^α_{β/μ}& =0\quadα,β,μ\text{ distinct},\\ z^α_{β/α}& =z^{α'}_{β/α'},\quad\forallα,α'≠β.}\tag{3A.38}$$


• Case ii :
  For a $3$-submanifold, $p=3,$ or special symmetric cases in higher dimensions, $p>3$ [E §15], it is possible to extend the conformal case by choosing a coordinate system such that $$ω^α_{/μ}=\e^{au+v_μ}δ^α_μ,\quad ω^{/μ}_α=\e^{-au-v_μ}δ_α^μ,\tag{3A.39}$$ We then have orthogonal coordinate line, and $$γ^γ_{αβ}-γ^γ_{βα}=0,\tag{3A.40}$$ $$\sqrt h=\e^{pau+\sum_μ v_μ},\tag{3A.41}$$ $$ω'^α_{/μ}=\e^{v_μ}δ^α_μ,\quad ω'^{/μ}_α=\e^{-v_μ}δ_α^μ,\tag{3A.42}$$ thus the equation $$\Big\{\e^{-v_μ}E^μ[∂_μ +\tfrac12∂_μ\sum_{ν≠μ}v_ν-A_μ] +M'\Big\}\varPsi=\boldsymbol0.\tag{3A.43}$$ This means that the space of parameters is no longer flat, its metric is $$h'_{μν}=\e^{2v_μ}δ_{μν},\tag{3A.44}$$ different from the metric of the submanifold $$g_{μν}=\e^{2u/(p-1)+2v_μ}δ_{μν},\tag{3A.45}$$ but it is in the same conformal class. Thus the potentials $v_μ$ define a conformal class of submanifolds. Like in the previous case we have (3A.36) and $$γ^α_{βα} =\e^{v_α-v_β-au}∂_β[u/(p-1)+v_α].\tag{3A.46}$$ Thus the additional constraints are $$\eqalign{z^α_{β/μ}& =0\quadα,β,μ\text{ distinct},\\ z^α_{β/α}& =\tfrac12\e^{v_α-v_β}∂_β[u/(p-1)+v_α],\quadα≠β.}\tag{3A.47}$$

3B. Conserved currents

The standard procedure is used to get the conserved current. We multiply the Dirac equation on the left by $E_P$ then by $\tilde\varPsi,$ and we take the reverse of the equation thus obtained : $$\begin{multline}\hspace6em \tilde\varPsi E_P\Big\{ω'^{/μ}_α E^α[∂_μ-\tfrac12a^a_{b/μ}E^{\ph b}_a] +\tfrac12∂_μ(ω'^{/μ}_α\sqrt h')/\sqrt h'E^α+\\ +\tfrac18(γ'^γ_{βα}-γ'^γ_{αβ})E^{\phαβ}_γ +m'_aE^a\Big\}\varPsi=\boldsymbol0. \hspace6em\tag{3B.1}\end{multline}$$ $$\begin{multline}\hspace6em \tilde\varPsi\Big\{ω'^{/μ}_α E^α[\lvec∂_μ+\tfrac12a^a_{b/μ}E^{\ph b}_a] +\tfrac12∂_μ(ω'^{/μ}_α\sqrt h')/\sqrt h'E^α+\\ -\tfrac14(γ'^γ_{βα}-γ'^γ_{αβ})E^{\phαβ}_γ +m'_aE^a\Big\}E_P\varPsi=\boldsymbol0. \hspace6em\tag{3B.$\tilde1$}\end{multline}$$ The tilde over $E_P$ is dropped because it is a mere change of sign. As $E^{\ph b}_a$ commutes with both $E^α$ and $E_P,$ either adding or subtracting these equations according to the parity of $p$ gives $$ω'^{/μ}_α∂_μ(\tilde\varPsi E_P E^α\varPsi) +∂_μ(ω'^{/μ}_α\sqrt h')/\sqrt h'(\tilde\varPsi E_P E^α\varPsi) =\boldsymbol0.\tag{3B.2}$$ Defining the tangent pseudo-vector current $$\boldsymbol J^{/μ}_P:=ω'^{/μ}_α\sqrt h'(\tilde\varPsi E_PE^α\varPsi),\tag{3B.3}$$ we have the continuity equation $$∂_μ\boldsymbol J^{/μ}_P=\boldsymbol0.\tag{3B.4}$$ This is the axial fermion-number current. But unlike the case of a surface, if $p>2$ these currents are not vectors and cannot be used to construct an immersion.

§4. Extended Gauß map

4A. Definition

As the local tangent space of the immersed submanifold is spanned by $\{\boldsymbol E_μ\},$ its normal vectors are the $\boldsymbol E_a.$ In the Clifford algebra, the normal space can be represented by a single multivector : $\boldsymbol E_{p+1}⋯\boldsymbol E_n.$ It is equivalent and more convenient to use its dual, the pseudo-scalar $\boldsymbol E_1⋯\boldsymbol E_p$ of the immersed manifold, since this avoids the reference to the ambient space. Then in the general case, the pseudo-scalar is $$\tilde\varPsi E_1\varPsi⋯\tilde\varPsi E_p\varPsi=ρ^{p-1}\tilde\varPsi E_P\varPsi.\tag{4A.1}$$ Since we are only interested in the direction, the norm is immaterial, so we shall take $$\boldsymbol\phi:=ρ^{-1}\tilde\varPsi E_P\varPsi =\varPsi^{-1}E_P\varPsi.\tag{4A.2}$$ We call this the extended Gauß map. From this expression we see that multiplying $\varPsi$ 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 ${\rm Spin}(n)$ $${{\rm Spin}(n)\over{\rm Spin}(p)×{\rm Spin}(q)} \sim{{\rm SO}(n)\over{\rm SO}(p)×{\rm SO}(q)},\tag{4A.3}$$ It is the real oriented Grassmann manifold $\tilde{\rm G}_{p,q},$ while ${\rm Lor}(p)×{\rm YM}(q)$ is the isotropy group. (Like $n,$ $p$ and $q$ are understood as $p_+,p_-$ and $q_+,q_-.$) Because of the invariance under ${\rm Lor}(p)×{\rm YM}(q),$ the Gauß map can be represented by $$\boldsymbol\phi=\exp\{-U/2\}E_P\exp\{U/2\}=R^{-1}E_P=E_PR\tag{4A.4}$$ 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 $$\displaylines{[{\frak lor},{\frak lor}]⊂{\frak lor},\quad [{\frak ym},{\frak ym}]⊂{\frak ym},\quad [{\frak lor},{\frak ym}]=\{\boldsymbol0\},\\ [{\frak lor},\Bbb W]⊂\Bbb W,\quad [{\frak ym},\Bbb W]⊂\Bbb W,\quad [\Bbb W,\Bbb W]⊂{\frak lor}\oplus{\frak ym}.}\tag{4A.8}$$

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 : $$\hspace-3em\displaylines{{\cal MRM}^{-1}= {\small\pmatrix{\cos(θ_1)& -\sin(θ_1)& 0& 0& ⋯\\ \sin(θ_1)& \phantom-\cos(θ_1)& 0& 0& ⋯\\ 0& 0& \cos(θ_2)& -\sin(θ_2)& ⋯\\ 0& 0& \sin(θ_2)& \phantom-\cos(θ_2)& ⋯\\ ⋮& ⋮& ⋮& ⋮& ⋱}}\\ ={\small\pmatrix{\cos(θ_1)& -\sin(θ_1)& 0& 0& ⋯\\ \sin(θ_1)& \phantom-\cos(θ_1)& 0& 0& ⋯\\ 0& 0& 1& 0& \dots\\0& 0& 0& 1& \dots\\ ⋮& ⋮& ⋮& ⋮& ⋱} \pmatrix{1& 0& 0& 0& ⋯\\0& 1& 0& 0& ⋯\\ 0& 0& \cos(θ_2)& -\sin(θ_2)& ⋯\\ 0& 0& \sin(θ_2)& \phantom-\cos(θ_2)& ⋯\\ ⋮& ⋮& ⋮& ⋮& ⋱}}⋯ =:{\cal R}'_1{\cal R}'_2⋯}\tag{4B.1}$$ Then $${\cal R=M}^{-1}{\cal R}'_1{\cal R}'_2⋯{\cal M =(M}^{-1}{\cal R}'_1{\cal M)(M}^{-1}{\cal R}'_2{\cal M})⋯ =:{\cal R}_1{\cal R}_2⋯\tag{4B.2}$$ 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 $$\begin{multline}\hspace{3em} {\cal W}={\small\pmatrix{0& -\sin(θ_1/2)& 0& 0& ⋯\\ \sin(θ_1/2)& 0& 0& 0& ⋯\\ 0& 0& 0& -\sin(θ_2/2)& ⋯\\ 0& 0& \sin(θ_2/2)& 0& ⋯\\ ⋮& ⋮& ⋮& ⋮& ⋱}}×\\ ×{\small\pmatrix{\cos(θ_1/2)& 0& 0& 0& ⋯\\ 0& \phantom-\cos(θ_1/2)& 0& 0& ⋯\\ 0& 0& \cos(θ_2/2)& 0& ⋯\\ 0& 0& 0& \phantom-\cos(θ_2/2)& ⋯\\ ⋮& ⋮& ⋮& ⋮& ⋱}}^{\Large -1}\\ =\small\pmatrix{0& -\tan(θ_1/2)& 0& 0& ⋯\\ \tan(θ_1/2)& 0& 0& 0& ⋯\\ 0& 0& 0& -\tan(θ_2/2)& ⋯\\ 0& 0& \tan(θ_2/2)& 0& ⋯\\ ⋮& ⋮& ⋮& ⋮& ⋱}. \hspace{3em}\tag{4B.5}\end{multline}$$ 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 $${\cal W} =\sum_i{{\cal R}_i^{1/2}-{\cal R}_i^{-1/2}\over{\cal R}_i^{1/2}+{\cal R}_i^{-1/2}} ={\sum_i({\cal R}_i^{1/2}-{\cal R}_i^{-1/2}) \prod_{j≠i}({\cal R}_j^{1/2}+{\cal R}_j^{-1/2}) \over\prod_k({\cal R}_k^{1/2}+{\cal R}_k^{-1/2})}.\tag{4B.8}$$ 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, explicitly with the matrices remarking 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 $$U=\sum_iU_i,\quad U_iU_i=±(θ_i)^2E,\quad[U_i,U_j]=\boldsymbol0,\tag{4B.12}$$ each corresponding to the $W_i,$ so that we get $$W=\sum_i W_i,\quad [W_i,W_j]=\boldsymbol0,\tag{4B.13}$$ $$W=\sum_i{\e^{U_i/2}-\e^{-U_i/2}\over\e^{U_i/2}+\e^{-U_i/2}} ={\sum_i[(\e^{U_i/2}-\e^{-U_i/2})\prod_{j≠i}(\e^{U_j/2}+\e^{-U_j/2})] \over\prod_k(\e^{U_k/2}+\e^{-U_k/2})}.\tag{4B.14}$$ The expansion of the numerator can be written as $$\begin{multline}\hspace6em n(\e^{U/2}-\e^{-U/2})+(n-2)\sum_{i=1}^n(\e^{U/2-U_i}-\e^{-U/2+U_i})+\\ +(n-4)\sum_{i\lt j}^n(\e^{U/2-U_i-U_j}-\e^{-U/2+U_i+U_j})+⋯ \hspace6em\tag{4B.15}\end{multline}$$ for the coefficients $(n-2k)\gt0,$ which arises from cancellation of identical terms because of the minus sign. And the denominator is $$\hspace-4em(\e^{U/2}+\e^{-U/2})+\sum_{i=1}^n(\e^{U/2-U_i}+\e^{-U/2+U_i}) +\sum_{i\lt j}^n(\e^{U/2-U_i-U_j}+\e^{-U/2+U_i+U_j})+⋯,\tag{4B.16}$$ with $[n/2]+1$ terms. Finally, by expanding the exponential functions all the products $U_iU_j⋯$ cancel out, leaving only the bivector or scalar terms, so we see that this is equal to the compact form $$W={⟨\e^{U/2}-\e^{-U/2}⟩_2\over⟨\e^{U/2}+\e^{-U/2}⟩} ={⟨\e^{U/2}⟩_2\over⟨\e^{U/2}⟩},\tag{4B.17}$$ 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 $∈fty_i,$ 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 transform $$\e^U=\prod_i\e^{U_i}=\prod_i{E+W_i\over E-W_i}={\e^{∧W}\over\e^{∧-W}},\tag{4B.19}$$ $$\e^{U/2}={\e^{∧W}\over|\e^{∧W}|},\tag{4B.20}$$ where $$\eqalign{ \e^{∧ W}:& =E+W+\tfrac12W∧ W+\tfrac1{3!}W∧ W∧ W+⋯\\ :& =⟨E⟩_0+⟨W⟩_2+\tfrac12⟨WW⟩_4+\tfrac1{3!}⟨WWW⟩_6+⋯\\ & =E+\sum_i W_i+\sum_{i\lt j}W_iW_j+\sum_{i\lt j\lt k}W_iW_jW_k+⋯, }\tag{4B.21}$$ $$|\e^{∧ W}|^2=\e^{∧ W}\e^{∧ -W}={\textstyle\prod_i}(E-W_iW_i).\tag{4B.22}$$ 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

There is an important problem. Given a prescribed Gauß map, how can we reconstruct the associated submanifold, that is, find the corresponding solution $\varPsi,$ its lift in the Clifford group? This naturally leads to the equation it must satisfy to be the Gauß map of some submanifold. This problem is completely solved for a surface in $\Bbb R^3$ [I], and in $\Bbb R^3$ and $\Bbb R^4$ [HO85]. Here we achieve only a partial resolution.

In the appropriate gauge, we can locally write $\varPsi$ simply as follow : $$\varPsi=:±\e^{u/2}\e^{V/2}\e^{U/2},\quad V=:v^{αβ}E_{αβ}.\tag{4C.1}$$ Then the Gauß map becomes $$\boldsymbol\phi=\varPsi^{-1}E_P\varPsi=E_P\e^U,\tag{4C.2}$$ so that we have the stereographic projection and back $$W={⟨(E_P^{-1}\boldsymbol\phi)^{1/2}⟩_2 \over⟨(E_P^{-1}\boldsymbol\phi)^{1/2}⟩},\quad \boldsymbol\phi=E_P{\e^{∧ W}\over\e^{∧-W}}.\tag{4C.3ab}$$ Now let us consider only normal spaces with an Euclidean signature. Although $V$ obviously cannot be expressed algebraically from $W$ like $U,$ it can differentially be so through the Dirac equation supposing that $\varPsi$ is a solution. We write it as $$\hspace-3em\Big\{ω'^{/μ}_α E^α[∂_μ-A_μ] +\tfrac12∂_μ(ω'^{/μ}_α\sqrt h')/\sqrt h'E^α +\tfrac18(γ'^γ_{βα}-γ'^γ_{αβ})E^{\phαβ}_γ +M'\Big\}\varPsi\varPsi^{-1}=\boldsymbol0.\tag{4C.4}$$ It is enough to only keep the terms with a single $E^a.$ The left hand side then reduces to $$ω'^{/μ}_α E^α∂_μ\varPsi\varPsi^{-1}+M'.\tag{4C.5}$$ As $$∂_μ\varPsi\varPsi^{-1} =\tfrac12∂_μ uE +(∂_μ\e^{V/2})\e^{-V/2} +\e^{V/2}(∂_μ\e^{U/2})\e^{-U/2}\e^{-V/2},\tag{4C.6}$$ only the derivative of $U$ remains : $$ω'^{/μ}_α E^α\e^{V/2}(∂_μ\e^{U/2})\e^{-U/2}\e^{-V/2}+M'.\tag{4C.7}$$ Considering the expansion [M, Lemma 5.4] $$(\d\e^X)\e^{-X}=\d X+\tfrac12[X,\d X]+\tfrac1{3!}[X,[X,\d X]]+⋯\tag{4C.8}$$ with the commutation rules (4A.8) and the fact that $U∈\Bbb W,$ all terms without $E^a$ are eliminated by writing $$\tfrac12ω'^{/μ}_α E^α\e^{V/2} [(∂_μ\e^{U/2})\e^{-U/2}-(∂_μ\e^{-U/2})\e^{U/2}]\e^{-V/2} +M'=\boldsymbol0.\tag{4C.9}$$ From the expression (4B.20) we calculate $$(∂_μ\e^{U/2})\e^{-U/2} ={(∂_μ\e^{∧W})\e^{∧-W}-|\e^{∧W}|∂_μ|\e^{∧W}|E \over|\e^{∧W}|^2},\tag{4C.10}$$ leading to the new equation $$\tfrac12ω'^{/μ}_α E^α\e^{V/2} [(∂_μ\e^{∧W})\e^{∧-W} -(∂_μ\e^{∧-W})\e^{∧W}]\e^{-V/2} =-|\e^{∧W}|^2M'.\tag{4C.11}$$ Now this contains vector and trivector terms. The latter are selected by subtracting this expression from its reverse, and we get the equation $$ω'^{/μ}_α \{E^α,\e^{V/2}[(∂_μ\e^{∧W})\e^{∧-W} -(∂_μ\e^{∧-W})\e^{∧W}]\e^{-V/2}\} =\boldsymbol0.\tag{4C.12}$$ It looks like trivial, but in general there is no simple commutation relation between any of the factors. Already for $p\geqslant3$ it is intractable. However, for a surface and any codimension (case 0,) it is simpler since $V$ anti-commutes with $E^α,$ $W,$ and its derivatives, and there are only products of an odd number of $W$’s because of the subtraction. This gives $$\begin{multline}\hspace6em \e^{2V} =[(E^μ∂_μ\e^{∧W})\e^{∧-W}-(E^μ∂_μ\e^{∧-W})\e^{∧W}]×\\ ×[\e^{∧W}(\e^{∧-W}\lvec∂_μ E^μ)-\e^{∧-W}(\e^{∧W}\lvec∂_μ E^μ)]^{-1}. \hspace6em\tag{4C.13}\end{multline}$$

To take into account the vector terms and get an additional equation, we multiply each side of (4C.11) by its reverse, giving $$\begin{multline}\hspace4em ω'^{/μ}_αE^α\e^{V/2} [(∂_μ\e^{∧W})\e^{∧-W}-(∂_μ\e^{∧-W})\e^{∧W}]×\\ ×[\e^{∧W}(\e^{∧-W}\lvec∂_ν)-\e^{∧-W}(\e^{∧W}\lvec∂_ν)] \e^{-V/2}ω'^{/ν}_βE^β =-4|\e^{∧W}|^4m'^2. \hspace4em\tag{4C.14}\end{multline}$$ Substituting $\e^{V/2},$ it is a constraint on $W.$ In the case 0 we can even eliminate it, and we have $$\begin{multline}\hspace6em [(E^μ∂_μ\e^{∧W})\e^{∧-W}-(E^μ∂_μ\e^{∧-W})\e^{∧W}]×\\ ×[\e^{∧W}(\e^{∧-W}\lvec∂_νE^ν)-\e^{∧-W}(\e^{∧W}\lvec∂_νE^ν)] =-4|\e^{∧W}|^4m'^2. \hspace6em\tag{4C.15}\end{multline}$$

Having $U$ and $V,$ we can inquire for $u.$ We have to keep only the tangent vectors in (4C.4), that is $$ω'^{/μ}_α E^α⟨∂_μ\varPsi\varPsi^{-1}⟩_∥ +\tfrac12∂_μ(ω'^{/μ}_α\sqrt h')/\sqrt h'E^α,\tag{4C.16}$$ where the symbol $∥$ means that we keep only the tangent terms, that is, in $E_{αβ}.$ Substituting (4C.6) and using (4C.10) we get $$\begin{multline}\hspace1em ω'^{/μ}_αE^α[\tfrac12∂_μuE +(∂_μ\e^{V/2})\e^{-V/2} +\e^{V/2}⟨[∂_μ\e^{∧-W},\e^{∧W}]⟩_∥\e^{-V/2}/|\e^{∧W}|^2]+\\ +\tfrac12∂_μ(ω'^{/μ}_α\sqrt h')/\sqrt h'E^α. \hspace5em\tag{4C.17}\end{multline}$$ Finally adding the reverse to get rid of the trivectors, the equation is $$[ω'^{/μ}_α ∂_μ u+∂_μ(ω'^{/μ}_α\sqrt h')/\sqrt h']E^α+\\ +ω'^{/μ}_α[E^α,(∂_μ\e^{V/2})\e^{-V/2} +\e^{V/2}⟨[∂_μ\e^{∧-W},\e^{∧W}]⟩_∥\e^{-V/2}/|\e^{∧W}|^2] =\boldsymbol0.\tag{4C.18}$$ For the case 0, it is $$E^\mu∂_\mu(uE+V) +2E^\mu⟨[∂_μ\e^{∧-W},\e^{∧W}]⟩_∥/|\e^{∧W}|^2 =\boldsymbol0.\tag{4C.19}$$ Under the condition that this system be compatible, $u$ is determined by integration up to an overall multiplicative constant. It is a necessary condition for $\varPsi$ satisfying the Dirac equation to exist.

§5. Known immersion formulæ and more

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 $$\{E^1∂_1+mE^2\}\varPsi=\boldsymbol0.\tag{5A.1}$$ It is put in its conventional form by multiplying by $E_2$ on the left $$\{E^{\ph1}_2∂_1-mE\}\varPsi=\boldsymbol0.\tag{5A.2}$$

Curve in $\Bbb R^2$

The very first case is a plane curve in $\Bbb R^2,$ then we need a representation of ${\rm C}ℓ_{2,0}$ : $$E=\boldsymbol 1,\quad E_1=iσ_1,\quad E_2=iσ_2,\quad E_{12}=-iσ_3,\tag{5A.3}$$ 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 $$\varPsi=ψ E-iψ^{12}σ_3 =ρ^{1/2}\exp\{+iϕ\tfrac12σ_3\},\tag{5A.5}$$ $$\tilde \varPsi=ψ E+iψ^{12}σ_3 =ρ^{1/2}\exp\{-iϕ\tfrac12σ_3\}.\tag{5A.$\tilde5$}$$ viz. $$\varPsi=ρ^{1/2}\pmatrix{\e^{iϕ/2}& 0\\0& \e^{-iϕ/2}},\quad \tilde\varPsi=ρ^{1/2}\pmatrix{\e^{-iϕ/2}& 0\\0& \e^{iϕ/2}}\tag{5A.6}$$ The Dirac equation (5A.2) is represented as $$\{iσ^3∂_1-m\boldsymbol1\}\varPsi=\boldsymbol0,\tag{5A.7}$$ and reduces to $$∂_1ρ^{1/2}=0,\quad ∂_1ϕ=2m.\tag{5A.8}$$ As $ρ$ is constant, with no loss of generality we can set it to $1.$ The tangent vector is $$\boldsymbol E_1=\tilde\varPsi E_1\varPsi=E_1\exp\{iϕσ_3\} =\pmatrix{0& i\e^{-iϕ}\\i\e^{iϕ}& 0} =\cos(ϕ)E_1+\sin(ϕ)E_2,\tag{5A.9}$$ then $2m$ is the curvature since it is the variation of the angle $ϕ$ between $\boldsymbol E_1$ and $E_1.$ Without going into the details, it is clear that $\boldsymbol E_1$ gives a direction at each $ξ^1,$ which is the arclength parameter, and we know that in one dimension it is always integrable. There is no difficulty in deriving the immersion formula.

Three dimensions

For $p=1$ and $q=2,$ the Dirac equation is $$\{E^1[E∂_1-\tfrac12aE^{\ph3}_2]+m_2E^2+m_3E^3\}\varPsi=\boldsymbol0.\tag{5A.10}$$ 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 $$\{E^{\ph1}_2[E∂_1-\tfrac12aE^{\ph3}_2]-mE\}\varPsi=\boldsymbol0.\tag{5A.11}$$

Curve in $\Bbb R^3$

For ${\rm C}ℓ_{3,0}$ 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 $$\displaylines{({\sf a⊗x})({\sf b⊗y})=({\sf ab})⊗({\sf xy}),\\ \bTr{\sf a⊗x}=\bTr{\sf a}\bTr{\sf x},\\ {\sf a}⊗({\sf x+y})={\sf a⊗x+a⊗y},\quad ({\sf a+b})⊗{\sf x=a⊗x+b⊗x},\\ (λ{\sf a})⊗{\sf x=a}⊗(λ{\sf x})=λ({\sf a⊗x}).}\tag{5A.13}$$ The spinor is given by $$\varPsi=ψ E+ψ^{jk}E_{jk} =\boldsymbol 1⊗(ψ-ψ^{jk}iσ_i)=:\boldsymbol 1⊗\varPhi\tag{5A.14}$$ $$\tilde\varPsi=ψ E-ψ^{jk}E_{jk}=\boldsymbol 1⊗\varPhi^†\tag{5A.$\tilde{14}$}$$ where $$\varPhi=:\pmatrix{φ& -\barχ\\χ& \phantom-\barφ}∈ \Bbb R×{\rm SU}(2),\tag{5A.15}$$ and $$\varPhi^†\varPhi=ρ\boldsymbol 1,\quad \tilde\varPsi\varPsi=ρ E.\tag{5A.16}$$ The Dirac equation (5A.11) is represented as $$\{\boldsymbol1⊗ iσ^3[\boldsymbol1⊗\boldsymbol1∂_1 +\tfrac12a\boldsymbol1⊗ iσ^1]-m\boldsymbol1⊗\boldsymbol1\}(\boldsymbol1⊗\varPhi)=0.\tag{5A.16’}$$ Dropping the redundant $\boldsymbol1⊗$ factor we have $$\{iσ^3[\boldsymbol1∂_1 +\tfrac12a iσ^1]-m\boldsymbol1\}\varPhi=\boldsymbol0.\tag{5A.17}$$ It directly gives the Maurer-Cartan form $$Z_1=ρ^{-1}∂_1\varPsi\tilde\varPsi=-\boldsymbol1⊗(\tfrac12a iσ_1+miσ_3).\tag{5A.18}$$ As there is no scalar term, $ρ$ is anew constant and set to $1.$ The tangent vector is $$\boldsymbol E_1 =\tilde\varPsi(σ_3⊗ iσ_1)\varPsi =(\boldsymbol1⊗\varPhi^†) (σ_3⊗ iσ_1)(\boldsymbol1⊗\varPhi) =σ_3⊗\varPhi^† iσ_1\varPhi.\tag{5A.19}$$ With this we calculate $$∂_1\boldsymbol E_1 =\tilde\varPsi[Z_1,σ_3⊗ iσ_1]\varPsi =2m\tilde\varPsi\boldsymbol(σ_3⊗ iσ_2)\varPsi=2m\boldsymbol E_2.\tag{5A.20}$$ 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 $$\{E^1∂_1+E^2∂_2+mE^3\}\varPsi=\boldsymbol0,\tag{5B.1}$$ and in the conventional form $$\{E^{\ph1}_3∂_1+E^{\ph2}_3∂_2-mE\}\varPsi=\boldsymbol0.\tag{5B.2}$$

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 $$\left\{-\boldsymbol1⊗ iσ^2∂_1+\boldsymbol1⊗ iσ^1∂_2 -m\boldsymbol1⊗\boldsymbol1\right\}(\boldsymbol1⊗\varPhi)=\boldsymbol0,\tag{5B.3}$$ and dropping the redundant $\boldsymbol1⊗$ factor we get the usual equation [I] $$\left\{-iσ^2∂_1+iσ^1∂_2-m\boldsymbol1\right\}\varPhi=\boldsymbol0.\tag{5B.4}$$ With the definition of complex variables and derivatives $$ζ:=ξ^1+iξ^2,\tag{5B.5}$$ $$∂:=∂_ζ=\tfrac12(∂_1-i∂_2),\quad \bar∂:=∂_\barζ=\tfrac12(∂_1+i∂_2),\tag{5B.6}$$ it splits into $$\left\{\eqalign{ \bar∂φ=\phantom-\tfrac12mχ,\\ ∂χ=-\tfrac12mφ. }\right.\tag{5B.7ab}$$ The treatment given in [I] provides all the results. Alternatively, for a surface in conformal coordinates (case 0,) the differential forms (3A.27) are $ω^α=\dξ^α.$ The general immersion formula is then $$\d\boldsymbol X=\boldsymbol E_α\dξ^α,\tag{5B.8}$$ that is $$\boldsymbol X(ξ)=x^i(ξ)E_i =\boldsymbol X(ξ_0)+∫_γ(\boldsymbol E_1\dξ^1+\boldsymbol E_2\dξ^2).\tag{5B.9}$$ where $γ$ stands for an arbitrary path form $ξ_0$ to $ξ.$ On introducing the step numbers $$E_±=E_1±iE_2=σ_3⊗ iσ_±,\quad σ_±=σ_1±iσ_2,\tag{5B.10}$$ it becomes $$\hspace-2em\boldsymbol X =\tfrac12(x^1+ix^2)E_-+\tfrac12(x^1-ix^2)E_++x^3E_3 =\boldsymbol X_0+\tfrac12∫_γ(\boldsymbol E_-\dζ+\boldsymbol E_+\d\barζ)\tag{5B.11}$$ with $$\boldsymbol E_±:=\tilde\varPsi E_±\varPsi=σ_3⊗\varPhi^† iσ_±\varPhi.\tag{5B.12}$$ Using the orthonormality identities $$⟨E^±E_∓⟩=-2E,\quad ⟨E^±E_±⟩=0,\quad ⟨E^±E_3⟩=0,\quad ⟨E^3E_3⟩=-E\tag{5B.13}$$ we get the components $$\hspace-3em\eqalign{x^1±ix^2 & =x^1_0+ix^2_0-\tfrac12⟨∫_γ(σ^3⊗ iσ^±) (σ^3⊗\varPhi^† iσ_-\varPhi\dζ'+σ^3⊗\varPhi^† iσ_+\varPhi\d\barζ')⟩\\ & =x^1_0+ix^2_0+∫_γ(\tfrac14\bTr{σ^±\varPhi^†σ_-\varPhi}\dζ' +\tfrac14\bTr{σ^±\varPhi^†σ_+\varPhi}\d\barζ'),}\tag{5B.14a}$$ $$\hspace-3em\eqalign{x^3 & =x^3_0-\tfrac12⟨∫_γ(σ^3⊗ iσ^3)(σ^3⊗\varPhi^† iσ_-\varPhi\dζ' +σ^3⊗\varPhi^† iσ_+\varPhi\d\barζ')⟩\\ & =x^3_0+∫_γ(\tfrac14\bTr{σ^3\varPhi^† σ_-\varPhi}\dζ' +\tfrac14\bTr{σ^3\varPhi^† σ_+\varPhi}\d\barζ').}\tag{5B.14b}$$ With the help of the general expressions $$\hspace-2em\bTr{σ^+\varPhi^†_mσ_-\varPhi_n} =\phantom-4φ_mφ_n,\quad \bTr{σ^+\varPhi^†_mσ_+\varPhi_n} =-4χ_mχ_n,\tag{5B.15a}$$ $$\hspace-2em\bTr{σ^-\varPhi^†_mσ_-\varPhi_n} =-4\barχ_m\barχ_n,\quad \bTr{σ^-\varPhi^†_mσ_+\varPhi_n} =\phantom-4\barφ_m\barφ_n,\tag{5B.15b}$$ $$\hspace-3em\bTr{σ^3\varPhi^†_mσ_-\varPhi_n} =2\barχ_mφ_n+2φ_m\barχ_n,\quad \bTr{σ^3\varPhi^†_mσ_+\varPhi_n} =2\barφ_mχ_n+2χ_m\barφ_n,\tag{5B.15c}$$ $$\hspace-2em\bTr{\varPhi^†_mσ_-\varPhi_n} =2\barχ_mφ_n-2φ_m\barχ_n,\quad \bTr{\varPhi^†_mσ_+\varPhi_n} =2\barφ_mχ_n-2χ_m\barφ_n,\tag{5B.15d}$$ it gives the known formula [K96, I].

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

For the representation of ${\rm C}ℓ_{2,1},$ complex $2×2$ 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 $$\varPsi=ψ E+ψ^{jk}E_{jk}=:\varXi,\tag{5B.20}$$ $$\tilde\varPsi=\varXi^+,\tag{5B.$\tilde{20}$}$$ with $$\varXi=\pmatrix{φ& \barχ\\χ& \barφ}∈\Bbb R×{\rm SU}(1,1),\quad \varXi^+=\pmatrix{\phantom-\barφ& -\barχ\\-χ& \phantom-φ}.\tag{5B.21}$$ The Dirac equation (5B.2) is represented as $$\{σ^2∂_1-σ^1∂_2-m\boldsymbol1\}\varXi=\boldsymbol0\tag{5B.22}$$ and splits into $$\left\{\eqalign{ \bar∂φ& =-\tfrac12imχ,\\ ∂χ& =\phantom-\tfrac12imφ. }\right.\tag{5B.23}$$ The step numbers and the tangent vectors are $$E_±=iσ_±,\tag{5B.24}$$ $$\boldsymbol E_±=\varXi^+ iσ_±\varXi.\tag{5B.25}$$ Using the formula (5B.11) and the orthonormality identities (5B.13) above, we get the components $$\hspace-1em\eqalign{x^1±ix^2 & =x^1_0±ix^2_0-\tfrac12⟨∫_γ iσ^± (\varXi^+ iσ_-\varXi\dζ'+\varXi^+ iσ_+\varXi\d\barζ')⟩\\ & =x^1_0±ix^2_0+∫_γ(\tfrac14\bTr{σ^±\varXi^+σ_-\varXi}\dζ' +\tfrac14\bTr{σ^±\varXi^+σ_+\varXi}\d\barζ'),}\tag{5B.26a}$$ $$\hspace-1em\eqalign{x^3 & =x^3_0+\tfrac12⟨∫_γσ^3(\varXi^+ iσ_-\varXi\dζ +\varXi^† iσ_+\varXi\d\barζ)⟩\\ & =x^3_0+i∫_γ(\tfrac14\bTr{σ^3\varXi^+σ_-\varXi}\dζ +\tfrac14\bTr{σ^3\varXi^+σ_+\varXi}\d\barζ).}\tag{5B.26b}$$ With the help of the general expressions $$\hspace-2em\bTr{σ^+\varXi_m^+σ_-\varXi_n} =\phantom-4φ_mφ_n,\quad \bTr{σ^+\varXi_m^+σ_+\varXi_n} =-4χ_mχ_n,\tag{5B.27a}$$ $$\hspace-2em\bTr{σ^-\varXi_m^+σ_-\varXi_n} =-4\barχ_m\barχ_n,\quad \bTr{σ^-\varXi_m^+σ_+\varXi_n} =\phantom-4\barφ_m\barφ_n,\tag{5B.27b}$$ $$\hspace-4em\bTr{σ^3\varXi_m^+σ_-\varXi_n} =-2\barχ_mφ_n-2φ_m\barχ_n,\quad \bTr{σ^3\varXi_m^+σ_+\varXi_n} =\phantom-2\barφ_mχ_n+2χ_m\barφ_n,\tag{5B.27c}$$ $$\hspace-3em\bTr{\varXi_m^+σ_-\varXi_n} =-2\barχ_mφ_n+2φ_m\barχ_n,\quad \bTr{\varXi_m^+σ_+\varXi_n} =\phantom-2\barφ_mχ_n-2χ_m\barφ_n,\tag{5B.27d}$$ it gives the final formula [K96, CT99] $$x^1+ix^2 =x^1_0+ix^2_0+∫_γ(φφ~\dζ' -χχ~\d\barζ'),\tag{5B.28a}$$ $$x^1-ix^2 =x^1_0-ix^2_0+∫_γ(-\barχ\barχ~\dζ' +\barφ\barφ~\d\barζ'),\tag{5B.28b}$$ $$x^3 =x^3_0+i∫_γ(-φ\barχ~\dζ' +\barφχ~\d\barζ').\tag{5B.28c}$$

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 ${\rm C}ℓ_{2,1}$ 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 $$\varPsi=ψ E+ψ^{jk}E_{jk}=:\varUpsilon.\tag{5B.33}$$ $$\tilde\varPsi=\varUpsilon^×\tag{5B.$\tilde{33}$}$$ with $$\varUpsilon=\pmatrix{a& b\\c& d}∈{\rm GL}(2\Bbb R),\quad \varUpsilon^×:=\pmatrix{\phantom-d& -b\\-c& \phantom-a}.\tag{5B.34}$$ The Dirac equation (5B.2) is represented as $$\{-iσ^2∂_1+σ^1∂_2-m\boldsymbol1\}\varUpsilon=\boldsymbol0.\tag{5B.35}$$ Since the matrix elements of $\varUpsilon$ 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 $$ζ^-=ξ^1-ξ^2,\quadζ^+=ξ^1+ξ^2,\tag{5B.37}$$ $$∂_-:=∂_{ζ^-}=\tfrac12(∂_1-∂_2),\quad ∂_+:=∂_{ζ^+}=\tfrac12(∂_1+∂_2).\tag{5B.38}$$ Then the equation splits into $$\left\{\eqalign{ ∂_-φ& =\phantom-\tfrac12mχ,\\ ∂_+χ& =-\tfrac12mφ. }\right.\tag{5B.39ab}$$ We also introduce other step numbers. $$E'_±=E_1±E_2=iσ_±,\quad (E'^±=E^1∓E^2=iσ^∓).\tag{5B.40}$$ The orthonormality identities are similar : $$⟨E'^±E'_∓⟩=-2E,\quad ⟨E'^±E'_±⟩=0,\quad ⟨E'^±E_3⟩=0,\quad ⟨E^3E_3⟩=-E,\tag{5B.41}$$ and the immersion formula is $$\hspace-3em\boldsymbol X =\tfrac12(x^1-x^2)E'_-+\tfrac12(x^1+x^2)E'_++x^3E_3 =\boldsymbol X_0 +\tfrac12∫_γ(\boldsymbol E'_-\dζ'^-+\boldsymbol E'_+\dζ'^+),\tag{5B.42}$$ with $$\boldsymbol E'_±=\varUpsilon^×iσ_±\varUpsilon.\tag{5B.43}$$ We get the components $$\eqalign{x^1±x^2 & =x^1_0±x^2_0-\tfrac12⟨∫_γ iσ^∓ [\varUpsilon^×iσ_-\varUpsilon\dζ'^-+\varUpsilon^×iσ_+\varUpsilon\dζ'^+]⟩\\ & =x^1_0±x^2_0+∫_γ(\tfrac14\bTr{σ^∓\varUpsilon^×σ_-\varUpsilon}\dζ'^- +\tfrac14\bTr{σ^∓\varUpsilon^×σ_+\varUpsilon}\dζ'^+),}\tag{5B.44a}$$ $$\eqalign{x^3 & =x^3_0-\tfrac12⟨∫_γ iσ^3[ \varUpsilon^×iσ_-\varUpsilon\dζ'^-+\varUpsilon^×iσ_+\varUpsilon\dζ'^+]⟩\\ & =x^3_0+∫_γ( \tfrac14\bTr{σ^3\varUpsilon^×σ_-\varUpsilon}\dζ'^- +\tfrac14\bTr{σ^3\varUpsilon^×σ_+\varUpsilon}\dζ'^+).}\tag{5B.44b}$$ With the help of the general expressions $$\hspace-2em\bTr{σ^+\varUpsilon^×_mσ_-\varUpsilon_n} =\phantom-4a_m a_n,\quad \bTr{σ^+\varUpsilon^×_mσ_+\varUpsilon_n} =-4c_mc_n,\tag{5B.45a}$$ $$\hspace-2em\bTr{σ^-\varUpsilon^×_mσ_-\varUpsilon_n} =-4b_mb_n,\quad \bTr{σ^-\varUpsilon^×_mσ_+\varUpsilon_n} =\phantom-4d_md_n,\tag{5B.45b}$$ $$\hspace-2em\bTr{σ^3\varUpsilon^×_mσ_-\varUpsilon_n} =-2a_mb_n-2b_ma_n,\quad \bTr{σ^3\varUpsilon^×_mσ_+\varUpsilon_n} =\phantom-2c_md_n+2d_mc_n,\tag{5B.45c}$$ $$\hspace-2em\bTr{\varUpsilon^×_mσ_-\varUpsilon_n} =\phantom-2a_mb_n-2b_ma_n,\quad \bTr{\varUpsilon^×_mσ_+\varUpsilon_n} =\phantom-2d_mc_n-2c_md_n,\tag{5B.45d}$$ it gives the final formula [L’08, P’16] $$x^1+x^2 =x^1_0+x^2_0+∫_γ(-b^2~\dζ'^-+d^2~\dζ'^+),\tag{5B.46a}$$ $$x^1-x^2 =x^1_0-x^2_0+∫_γ(a^2~\dζ'^--c^2~\dζ'^+),\tag{5B.46b}$$ $$x^3 =x^3_0+∫_γ(-ab~\dζ'^+-cd~\dζ'^+).\tag{5B.46c}$$

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 $$\{E^1∂_1+E^2∂_2+m_3E^3+m_4E^4\}\varPsi=\boldsymbol0\tag{5B.47}$$ which multiplied on the left by $E_3$ is $$\{E^{\ph1}_3∂_1+E^{\ph2}_3∂_2-m_3+m_4E^{\ph4}_3\}\varPsi=\boldsymbol0\tag{5B.48}$$

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

We take as a representation of ${\rm C}ℓ_{4,0}$ : $$E=\boldsymbol1⊗\boldsymbol1,\quad \eqalign{E_i& =iσ_2⊗σ_i\\E_4& =σ_1⊗ i\boldsymbol1},\quad \eqalign{E_{jk}& =-\boldsymbol1⊗ iσ_i\\E_{4i}& =-σ_3⊗ iσ_i},\\ \eqalign{E_{4jk}& =σ_1⊗σ_i\\E_{123}& =σ_2⊗\boldsymbol1},\quad E_{1234}=σ_3⊗\boldsymbol1.\tag{5B.49}$$ The spinor is given by $$\eqalign{ \varPsi& =(ψ E+ψ^{jk}E_{jk})+(ψ^{1234}E_{1234}+ψ^{4i}E_{4i})\\ & =\boldsymbol1⊗(ψ\boldsymbol1-iψ^{jk}σ_i) +σ_3⊗(ψ^{1234}\boldsymbol1-iψ^{4i}σ_i)\\ & =:\boldsymbol1⊗\tfrac12(\varPhi_1+\varPhi_2)+σ_3⊗\tfrac12(\varPhi_1-\varPhi_2),}\tag{5B.50}$$ $$\tilde\varPsi =\boldsymbol1⊗\tfrac12(\varPhi_1^†+\varPhi_2^†) +σ_3⊗\tfrac12(\varPhi_1^†-\varPhi_2^†),\tag{5B.$\tilde{50}$}$$ where $$\varPhi_1,\varPhi_2∈\Bbb R×{\rm SU}(2).\tag{5B.51}$$ It is not really an element of the special Clifford group unless $\varPhi_1^†\varPhi_1=\varPhi_2^†\varPhi_2.$ But we can write $$\varPsi=\e^{u^NE_N}\varPsi',\tag{5B.52}$$ where $\varPsi'$ has this property, then $$\tilde\varPsi E_μ\varPsi=\tilde\varPsi'E_μ\varPsi'$$ since $E_μ$ anticommute with $E_N$ and $\tilde E_N=E_N,$ so that it amounts to the same and is simpler. The Dirac equation (5B.48) is represented as $$\begin{multline}\hspace6em \{-\boldsymbol1⊗ iσ^2∂_1 +\boldsymbol1⊗ iσ^1∂_2 -m_3\boldsymbol1⊗\boldsymbol1 +m_4σ^3⊗ iσ^3\}×\\ ×[\boldsymbol1⊗\tfrac12(\varPhi_1+\varPhi_2) +σ_3⊗\tfrac12(\varPhi_1-\varPhi_2)]=0, \hspace6em\tag{5B.53}\end{multline}$$ and splits into $$\eqalign{\{-iσ^2∂_1+iσ^1∂_2-m_3\boldsymbol1+m_4iσ^3\}\varPhi_1& =\boldsymbol0,\\ \{-iσ^2∂_1+iσ^1∂_2-m_3\boldsymbol1-m_4iσ^3\}\varPhi_2& =\boldsymbol0.}\tag{5B.54ab}$$ 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 $$E_±=E_1±iE_2=σ_2⊗ iσ_±,\tag{5B.57}$$ $$\boldsymbol E_± =σ_+⊗\tfrac12\varPhi_1^†σ_±\varPhi_2 -σ_-⊗\tfrac12\varPhi_2^†σ_±\varPhi_1.\tag{5B.58}$$ With the orthonormality identities $$\displaylines{⟨E^±E_∓⟩=-2E,\quad⟨E^±E_±⟩=0,\quad⟨E^±E_3⟩=0,\quad⟨E^±E_4⟩=0,\\ ⟨E^3E_3⟩=-E,\quad⟨E^3E_4⟩=0,\quad⟨E^4E_4⟩=-E,}\tag{5B.59}$$ and the immersion formula $$\eqalign{ \boldsymbol X & =\tfrac12(x^1+ix^2)E_-+\tfrac12(x^1-ix^2)E_++x^3E_3+x^4E_4\\ & =\boldsymbol X_0+\tfrac12∫_γ(\boldsymbol E_-\dζ +\boldsymbol E_+\d\barζ),}\tag{5B.60}$$ we get the components $$\hspace-5em\eqalign{ x^1±ix^2 & =x^1_0±ix^2_0 -\tfrac12⟨∫_γ(σ^2⊗ iσ^±) [(σ_+⊗\tfrac12\varPhi_1^†σ_-\varPhi_2-σ_ -⊗\tfrac12\varPhi_2^†σ_-\varPhi_1)\dζ'+\\ & \hspace5em+(σ_+⊗\tfrac12\varPhi_1^†σ_+\varPhi_2-σ_ -⊗\tfrac12\varPhi_2^†σ_+\varPhi_1)\d\barζ']⟩\\ & =x^1_0±ix^2_0 +∫_γ(\tfrac18\bTr{σ^±\varPhi_1^†σ_-\varPhi_2 +σ^±\varPhi_2^†σ_-\varPhi_1}\dζ'+\\ & \hspace5em+\tfrac18\bTr{σ^±\varPhi_1^†σ_+\varPhi_2 +σ^±\varPhi_2^†σ_+\varPhi_1}\d\barζ'),}\tag{5B.61a}$$ $$\hspace-7em\eqalign{x^3 & =x^3_0-\tfrac12⟨∫_γ(σ^2⊗ iσ^3)[ (σ_+⊗\tfrac12\varPhi_1^†σ_-\varPhi_2 -σ_-⊗\tfrac12\varPhi_2^†σ_-\varPhi_1)\dζ'+\\ & \hspace5em+(σ_+⊗\tfrac12\varPhi_1^†σ_+\varPhi_2 -σ_-⊗\tfrac12\varPhi_2^†σ_+\varPhi_1)\d\barζ']⟩\\ & =x^3_0+∫_γ(\tfrac18\bTr{σ^3\varPhi_1^†σ_-\varPhi_2 +σ^3\varPhi_2^†σ_-\varPhi_1}\dζ'+\\ & \hspace5em+\tfrac18\bTr{σ^3\varPhi_1^†σ_+\varPhi_2 +σ^3\varPhi_2^†σ_+\varPhi_1}\d\barζ'),}\tag{5B.61b}$$ $$\hspace-3em\eqalign{x^4 & =x^4_0-\tfrac12⟨∫_γ(σ^1⊗ i\boldsymbol1) [(σ_+⊗\tfrac12\varPhi_1^†σ_+\varPhi_2 -σ_-⊗\tfrac12\varPhi_2^†σ_+\varPhi_1)\dζ'+\\ & \hspace5em+(σ_+⊗\tfrac12\varPhi_1^†σ_-\varPhi_2 -σ_-⊗\tfrac12\varPhi_2^†σ_-\varPhi_1)\d\barζ']⟩\\ & =x^4_0-i∫_γ( \tfrac18\bTr{\varPhi_1^†σ_+\varPhi_2 -\varPhi_2^†σ_+\varPhi_1}\dζ' +\tfrac18\bTr{\varPhi_1^†σ_-\varPhi_2 -\varPhi_2^†σ_-\varPhi_1}\d\barζ').}\tag{5B.61c}$$ With the help of the general expressions (5B.15) it gives the final formula [K96] $$x^1+ix^2=x^1_0+ix^2_0+∫_γ(φ_1φ_2~\dζ' -χ_1χ_2~\d\barζ'),\tag{5B.62a}$$ $$x^1-ix^2=x^1_0-ix^2_0+∫_γ(-\barχ_1\barχ_2~\dζ' +\barφ_1\barφ_2~\d\barζ')\tag{5B.62b}$$ $$x^3+ix^4=x^3_0+ix^4_0+∫_γ(\barχ_1φ_2~\dζ' +\barφ_1χ_2~\d\barζ'),\tag{5B.62c}$$ $$x^3-ix^4=x^3_0-ix^4_0+∫_γ(φ_1\barχ_2~\dζ' +χ_1\barφ_2~\d\barζ').\tag{5B.62d}$$

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

Using the symbols (5B.16), a representation of ${\rm C}ℓ_{2,2}$ 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 $$\eqalign{\varPsi& =ψ E+ψ^{jk}E_{jk}+ψ^{4i}E_{4i}+ψ^{1234}E_{1234}\\ & =\boldsymbol1⊗(ψ\boldsymbol1+ψ^{jk}\varsigma_i^†) +σ_3⊗(ψ^{1234}\boldsymbol1-ψ^{4i}\varsigma_i)\\ & =:\boldsymbol1⊗\tfrac12(\varXi_1+\varXi_2) +σ_3⊗\tfrac12(\varXi_1-\varXi_2),}\tag{5B.65}$$ $$\eqalign{\tilde\varPsi& =ψ E-ψ^{jk}E_{jk}-ψ^{4i}E_{4i}+ψ^{1234}E_{1234}\\ & =:\boldsymbol1⊗\tfrac12(\varXi_1^++\varXi_2^+) +σ_3⊗\tfrac12(\varXi_1^+-\varXi_2^+),}\tag{5B.$\tilde{65}$}$$ and $\varXi∈\Bbb R×{\rm SU}(1,1)$ is defined by (5B.21). The Dirac equation (5B.48) is represented as $$\begin{multline}\hspace6em \{\boldsymbol1⊗σ^2∂_1 -\boldsymbol1⊗σ^1∂_2 -m_3\boldsymbol1⊗\boldsymbol1-m_4σ_3⊗ iσ^3\}×\\ ×[\boldsymbol1⊗\tfrac12(\varXi_1+\varXi_2) +σ_3⊗\tfrac12(\varXi_1-\varXi_2)]=\boldsymbol0 \hspace6em\tag{5B.66}\end{multline}$$ and splits into $$\eqalign{ \{σ^2∂_1-σ^1∂_2-m_3\boldsymbol1-m_4iσ^3\}\varXi_1& =\boldsymbol0,\\ \{σ^2∂_1-σ^1∂_2-m_3\boldsymbol1+m_4iσ^3\}\varXi_2& =\boldsymbol0.}\tag{5B.67ab}$$ 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 $$E_±=E_1±iE_2=σ_2⊗ iσ_±,\tag{5B.69}$$ $$\boldsymbol E_± =σ_+⊗\tfrac12\varXi^+_1σ_±\varXi_2 -σ_-⊗\tfrac12\varXi^+_2σ_±\varXi_1.\tag{5B.70}$$ From the immersion formula (5B.60) and the orthonormality identities (5B.59) we get the components $$\hspace-5em\eqalign{x^1±ix^2 & =x^1_0±ix^2_0-\tfrac12⟨∫_γ (σ^2⊗ iσ^±)[(σ_+⊗\tfrac12\varXi^+_1σ_-\varXi_2 -σ_-⊗\tfrac12\varXi^+_2σ_-\varXi_1)\dζ'+\\ & \hspace5em+(σ_+⊗\tfrac12\varXi^+_1σ_+\varXi_2 -σ_-⊗\tfrac12\varXi^+_2σ_+\varXi_1)\d\barζ']⟩\\ & =x^1_0±ix^2_0+∫_γ (\tfrac18\bTr{σ^±\varXi^+_1σ_-\varXi_2+σ^±\varXi^+_2σ_-\varXi_1}\dζ'+\\ & \hspace5em+\tfrac18\bTr{σ^±\varXi^+_1σ_+\varXi_2+σ^±\varXi^+_2σ_+\varXi_1}\d\barζ'),}\tag{5B.71a}$$ $$\hspace-3em\eqalign{x^3 & =x^3_0-\tfrac12⟨∫_γ (σ^2⊗σ^3)[ (σ_+⊗\tfrac12\varXi^+_1σ_-\varXi_2 -σ_-⊗\tfrac12\varXi^+_2σ_-\varXi_1)\dζ'+\\ & \hspace2em+(σ_+⊗\tfrac12\varXi^+_1σ_+\varXi_2 -σ_-⊗\tfrac12\varXi^+_2σ_+\varXi_1)\d\barζ']⟩\\ & =x^3_0+i∫_γ (\tfrac18\bTr{σ^3\varXi^+_1σ_-\varXi_2+σ^3\varXi^+_2σ_-\varXi_1}\dζ'+\\ & \hspace2em+\tfrac18\bTr{σ^3\varXi^+_1σ_+\varXi_2+σ^3\varXi^+_2σ_+\varXi_1}\d\barζ'),}\tag{5B.71b}$$ $$\hspace-2em\eqalign{x^4 & =x^4_0-\tfrac12⟨∫_γ (-σ^1⊗\boldsymbol1)[ (σ_+⊗\tfrac12\varXi^+_1σ_-\varXi_2 -σ_-⊗\tfrac12\varXi^+_2σ_-\varXi_1)\dζ'+\\ & \hspace2em+(σ_+⊗\tfrac12\varXi^+_1σ_+\varXi_2 -σ_-⊗\tfrac12\varXi^+_2σ_+\varXi_1)\d\barζ']⟩\\ & =x^4_0+∫_γ (\tfrac18\bTr{\varXi^+_1σ_-\varXi_2-\varXi^+_2σ_-\varXi_1}\dζ'+\\ & \hspace2em+\tfrac18\bTr{\varXi^+_1σ_+\varXi_2-\varXi^+_2σ_+\varXi_1}\d\barζ').}\tag{5B.71c}$$ With the help of the general expressions (5B.27) it gives the final formula [K96] $$x^1+ix^2 =x^1_0+ix^2_0+∫_γ(φ_1φ_2~\dζ'-χ_1χ_2~\d\barζ'),\tag{5B.72a}$$ $$x^1-ix^2 =x^1_0-ix^2_0+∫_γ(-\barχ_1\barχ_2\dζ'+\barφ_1\barφ_2\d\barζ'),\tag{5B.72b}$$ $$x^3+ix^4 =x^3_0+ix^4_0+i∫_γ(-\barχ_1φ_2~\dζ'+\barφ_1χ_2~\d\barζ'),\tag{5B.72c}$$ $$x^3-ix^4 =x^3_0-ix^4_0+i∫_γ(-φ_1\barχ_2~\dζ'+χ_1\barφ_2~\d\barζ').\tag{5B.72d}$$

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

Using the symbols (5B.29), another representation of ${\rm C}ℓ_{2,2}$ 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 $$\eqalign{\varPsi& =ψ E+ψ^{jk}E_{jk}+ψ^{4i}E_{4i}+ψ^{1234}E_{1234}\\ & =\boldsymbol1⊗(ψ\boldsymbol1+ψ^{jk}{\sf s}^t_i) +σ_3⊗(ψ^{1234}\boldsymbol1-ψ^{4i}{\sf s}_i)\\ & =:\boldsymbol1⊗\tfrac12(\varUpsilon_1+\varUpsilon_2) +σ_3⊗\tfrac12(\varUpsilon_1-\varUpsilon_2),}\tag{5B.75}$$ $$\eqalign{\tilde\varPsi& =ψ E-ψ^{jk}E_{jk}-ψ^{4i}E_{4i}+ψ^{1234}E_{1234}\\ & =:\boldsymbol1⊗\tfrac12(\varUpsilon_1^×+\varUpsilon_2^×) +σ_3⊗\tfrac12(\varUpsilon_1^×-\varUpsilon_2^×),}\tag{5B.$\tilde{75}$}$$ where $\varUpsilon∈{\rm GL}(2\Bbb R)$ is defined in (5B.34). The Dirac equation (5B.48) is represented as $$\{-\boldsymbol1⊗iσ^2∂_1+\boldsymbol1⊗σ^1∂_2 -m_3\boldsymbol1⊗\boldsymbol1-m_4σ^3⊗σ^3\}\varPsi=\boldsymbol0.\tag{5B.76}$$ It splits into $$\eqalign{ \{-iσ^2∂_1+σ^1∂_2 -m_3\boldsymbol1-m_4σ^3\}\varUpsilon_1& =\boldsymbol0,\\ \{-iσ^2∂_1+σ^1∂_2 -m_3\boldsymbol1+m_4σ^3\}\varUpsilon_2& =\boldsymbol0,}\tag{5B.77ab}$$ 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 $$E'_±=E_1±E_2=iσ_2⊗σ_±\quad(E'^±=E^1∓E^2=iσ^2⊗σ^∓),\tag{5B.79}$$ $$\boldsymbol E'_± =σ_+⊗\tfrac12\varUpsilon_1^×σ_±\varUpsilon_2 -σ_-⊗\tfrac12\varUpsilon_2^×σ_±\varUpsilon_1.\tag{5B.80}$$ From the immersion formula $$\eqalign{ \boldsymbol X & =\tfrac12(x^1-x^2)E'_-+\tfrac12(x^1+x^2)E'_++x^3E_3+x^4E_4\\ & =\boldsymbol X_0 +\tfrac12∫_γ(\boldsymbol E'_-\dζ'^-+\boldsymbol E'_+\dζ'^+) }\tag{5B.81}$$ and the orthonormality identities (5B.59), we get the components $$\hspace-4em\eqalign{x^1±x^2 & =x^1_0±x^2_0-\tfrac12⟨∫_γ (iσ^2⊗σ^∓) [(σ_+⊗\tfrac12\varUpsilon_1^×σ_-\varUpsilon_2 -σ_-⊗\tfrac12\varUpsilon_2^×σ_-\varUpsilon_1)\dζ'^-+\\ & \hspace5em+(σ_+⊗\tfrac12\varUpsilon_1^×σ_+\varUpsilon_2 -σ_-⊗\tfrac12\varUpsilon_2^×σ_+\varUpsilon_1)\dζ'^+]⟩\\ & =x^1_0±x^2_0+∫_γ (\tfrac18\bTr{σ^∓\varUpsilon_1^×σ_-\varUpsilon_2 +σ^∓\varUpsilon_2^×σ_-\varUpsilon_1}\dζ'^-+\\ & \hspace5em+\tfrac18\bTr{σ^∓\varUpsilon_1^×σ_+\varUpsilon_2 +σ^∓\varUpsilon_2^×σ_+\varUpsilon_1}\dζ'^+),}\tag{5B.82a}$$ $$\hspace-4em\eqalign{x^3 & =x^3_0-\tfrac12⟨∫_γ(iσ^2⊗σ^3) [(σ_+⊗\tfrac12\varUpsilon_1^×σ_-\varUpsilon_2 -σ_-⊗\tfrac12\varUpsilon_2^×σ_-\varUpsilon_1)\dζ'^-+\\ & \hspace3em+(σ_+⊗\tfrac12\varUpsilon_1^×σ_+\varUpsilon_2 -σ_-⊗\tfrac12\varUpsilon_2^×σ_+\varUpsilon_1)\dζ'^+]⟩\\ & =x^3_0+∫_γ(\tfrac18\bTr{σ^3\varUpsilon_1^×σ_-\varUpsilon_2 +σ^3\varUpsilon_2^×σ_-\varUpsilon_1}\dζ'^-+\\ & \hspace3em+\tfrac18\bTr{σ^3\varUpsilon_1^×σ_+\varUpsilon_2 +σ^3\varUpsilon_2^×σ_+\varUpsilon_1}\dζ'^+),}\tag{5B.82b}$$ $$\hspace-2em\eqalign{x^4 & =x^4_0-\tfrac12∫_γ(-σ_1⊗\boldsymbol1) ([σ_+⊗\tfrac12\varUpsilon_1^×σ_-\varUpsilon_2 -σ_-⊗\tfrac12\varUpsilon_2^×σ_-\varUpsilon_1]\dζ'^-+\\ & \hspace3em+[σ_+⊗\tfrac12\varUpsilon_1^×σ_+\varUpsilon_2 -σ_-⊗\tfrac12\varUpsilon_2^×σ_+\varUpsilon_1]\dζ'^+).\\ & =x^4_0+\tfrac12∫_γ (\tfrac18\bTr{\varUpsilon_1^×σ_-\varUpsilon_2 -\varUpsilon_2^×σ_-\varUpsilon_1}\dζ'^-+\\ & \hspace3em+\tfrac18\bTr{\varUpsilon_1^×σ_+\varUpsilon_2 -\varUpsilon_2^×σ_+\varUpsilon_1}\dζ'^+).}\tag{5B.82c}$$ With the help of the general expressions (5B.45), it gives the final formula [K96, P’16] $$x^1+x^2 =x^1_0+x^2_0+∫_γ(-b_1b_2~\dζ'^-+d_1d_2~\dζ'^+),\tag{5B.83a}$$ $$x^1-x^2 =x^1_0-x^2_0+∫_γ(a_1a_2~\dζ'^--c_1c_2~\dζ'^+),\tag{5B.83b}$$ $$x^3+x^4=x^3_0+x^4_0+∫_γ(-b_1a_2~\dζ'^-+d_1c_2~\dζ'^+),\tag{5B.83c}$$ $$x^3-x^4=x^3_0-x^4_0+∫_γ(-a_1b_2~\dζ'^-+c_1d_2~\dζ'^+).\tag{5B.83d}$$

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

A representation of ${\rm C}ℓ_{3,1}$ 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 $$\eqalign{ \varPsi& =\boldsymbol1⊗(ψ-ψ^{jk}iσ_i)+iσ_3⊗(ψ^{1234}-ψ^{4i}iσ_i) =:\boldsymbol1⊗\varPhi'+iσ_3⊗\varPhi''\\ & =:\boldsymbol1⊗\tfrac12(\varUpsilon_{12}+\varUpsilon_{21}) +σ_3⊗\tfrac12(\varUpsilon_{12}-\varUpsilon_{21}),}\tag{5B.86}$$ $$\tilde\varPsi=:\boldsymbol1⊗\varPhi'^†+iσ_3⊗\varPhi''^† =\boldsymbol1⊗\tfrac12(\varUpsilon^†_{12}+\varUpsilon^†_{21}) -σ_3⊗\tfrac12(\varUpsilon^†_{12}-\varUpsilon^†_{21}),\tag{5B.$\tilde{86}$}$$ where $\varUpsilon_{12},\varUpsilon_{21}∈{\rm GL}(2\Bbb C)$ are defined as $$\varUpsilon_{12}=\varPhi'+i\varPhi''=\pmatrix{φ'+iφ''& -\barχ'-i\barχ''\\ χ'+iχ''& \phantom-\barφ'+i\barφ''} =:\pmatrix{φ_1& -\barχ_2\\χ_1& \phantom-\barφ_2},\tag{5B.87a}$$ $$\varUpsilon_{21}=\varPhi'-i\varPhi''=\pmatrix{φ'-iφ''& -\barχ'+i\barχ''\\ χ'-iχ''& \phantom-\barφ'-i\barφ''} =\pmatrix{φ_2& -\barχ_1\\χ_2& \phantom-\barφ_1}.\tag{5B.87b}$$ The Dirac equation (5B.48) is represented as $$\begin{multline}\hspace6em \{-\boldsymbol1⊗ iσ^2∂_1 +\boldsymbol1⊗ iσ^1∂_2 -m_3\boldsymbol1⊗\boldsymbol1 -m_4σ^3⊗σ^3\}×\\ ×[\boldsymbol1⊗\tfrac12(\varUpsilon_{12}+\varUpsilon_{21}) +σ_3⊗\tfrac12(\varUpsilon_{12}-\varUpsilon_{21})]=\boldsymbol0. \hspace6em\tag{5B.88}\end{multline}$$ It splits into $$\{-iσ^2∂_1+iσ^1∂_2 -m_3+m_4σ^3\}\varUpsilon_{12}=\boldsymbol0,\\ \{-iσ^2∂_1+iσ^1∂_2 -m_3-m_4σ^3\}\varUpsilon_{21}=\boldsymbol0,\tag{5B.89ab}$$ 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 $$E_±=E_1±iE_2=iσ_2⊗σ_±,\tag{5B.91}$$ $$\boldsymbol E_± =σ_+⊗\tfrac12\varUpsilon^†_{21}σ_±\varUpsilon_{21} -σ_-⊗\tfrac12\varUpsilon^†_{12}σ_±\varUpsilon_{12}.\tag{5B.92}$$ From the immersion formula (5B.60) and the orthonormality identities (5B.59) we get the components $$\hspace-4.5em\eqalign{x^1±ix^2 & =x^1_0±ix^2_0-\tfrac12⟨∫_γ (iσ^2⊗σ^±) [(σ_+⊗\tfrac12\varUpsilon^†_{21}σ_-\varUpsilon_{21} -σ_-⊗\tfrac12\varUpsilon^†_{12}σ_-\varUpsilon_{12})\dζ'+\\ & \hspace5em+(σ_+⊗\tfrac12\varUpsilon^†_{21}σ_+\varUpsilon_{21} -σ_-⊗\tfrac12\varUpsilon^†_{12}σ_+\varUpsilon_{12})\d\barζ']⟩\\ & =x^1_0±ix^2_0+∫_γ (\tfrac18\bTr{σ^±\varUpsilon^†_{12}σ_-\varUpsilon_{12} +σ^±\varUpsilon^†_{21}σ_-\varUpsilon_{21}}\dζ'+\\ & \hspace5em+\tfrac18\bTr{σ^±\varUpsilon^†_{12}σ_+\varUpsilon_{12} +σ^±\varUpsilon^†_{21}σ_+\varUpsilon_{21}}\d\barζ'),}\tag{5B.93a}$$ $$\hspace-4.5em\eqalign{x^3 & =x^3_0-\tfrac12⟨∫_γ (iσ^2⊗σ^3) [(σ_+⊗\tfrac12\varUpsilon^†_{21}σ_-\varUpsilon_{21} -σ_-⊗\tfrac12\varUpsilon^†_{12}σ_-\varUpsilon_{12})\dζ'+\\ & \hspace3em+(σ_+⊗\tfrac12\varUpsilon^†_{21}σ_+\varUpsilon_{21} -σ_-⊗\tfrac12\varUpsilon^†_{12}σ_+\varUpsilon_{12})\d\barζ']⟩\\ & =x^3_0+∫_γ (\tfrac18\bTr{σ^3\varUpsilon^†_{12}σ_-\varUpsilon_{12} +σ^3\varUpsilon^†_{21}σ_-\varUpsilon_{21}}\dζ'+\\ & \hspace3em+\tfrac18\bTr{σ^3\varUpsilon^†_{12}σ_+\varUpsilon_{12} +σ^3\varUpsilon^†_{21}σ_+\varUpsilon_{21}}\d\barζ'),}\tag{5B.93b}$$ $$\hspace-4.5em\eqalign{x^4 & =x^4_0-\tfrac12⟨∫_γ (σ^1⊗\boldsymbol1) [(σ_+⊗\tfrac12\varUpsilon^†_{21}σ_-\varUpsilon_{21} -σ_-⊗\tfrac12\varUpsilon^†_{12}σ_-\varUpsilon_{12})\dζ'+\\ & \hspace3em+(σ_+⊗\tfrac12\varUpsilon^†_{21}σ_+\varUpsilon_{21} -σ_-⊗\tfrac12\varUpsilon^†_{12}σ_+\varUpsilon_{12})\d\barζ']⟩\\ & =x^4_0+∫_γ (\tfrac18\bTr{\varUpsilon^†_{12}σ_-\varUpsilon_{12} -\varUpsilon^†_{21}σ_-\varUpsilon_{21}}\dζ'+\\ & \hspace3em+\tfrac18\bTr{\varUpsilon^†_{12}σ_+\varUpsilon_{12} -\varUpsilon^†_{21}σ_+\varUpsilon_{21}}\d\barζ').}\tag{5B.93c}$$ The evaluation of the traces gives the final formula [K96] $$x^1+ix^2 =x^1_0+ix^2_0+∫_γ (φ_1φ_2~\dζ'-χ_1χ_2~\d\barζ'),\tag{5B.94a}$$ $$x^1-ix^2 =x^1_0-ix^2_0+∫_γ (-\barχ_1\barχ_2~\dζ'+\barφ_1\barφ_2\d\barζ'),\tag{5B.94b}$$ $$x^3+x^4 =x^3_0+x^4_0+∫_γ (\barχ_1φ_1~\dζ'+\barφ_1χ_1~\d\barζ'),\tag{5B.94c}$$ $$x^3-x^4 =x^3_0-x^4_0+∫_γ (φ_2\barχ_2~\dζ'+χ_2\barφ_2~\d\barζ').\tag{5B.94d}$$

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

Using the symbols defined in (5B.29), another representation of ${\rm C}ℓ_{3,1}$ 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 $$\eqalign{\varPsi& =\boldsymbol1⊗(ψ\boldsymbol1+ψ^{jk}{\sf s}^t_i) +σ_3⊗i(ψ^{1234}\boldsymbol1-ψ^{4i}{\sf s}_i) =:\boldsymbol1⊗\varUpsilon'+σ_3⊗ i\varUpsilon''\\ & =:\boldsymbol1⊗\tfrac12(\varUpsilon+\bar\varUpsilon) +σ_3⊗\tfrac12(\varUpsilon-\bar\varUpsilon),}\tag{5B.97}$$ $$\tilde\varPsi=\boldsymbol1⊗\tfrac12(\varUpsilon^×+\bar\varUpsilon^×) +σ_3⊗\tfrac12(\varUpsilon^×-\bar\varUpsilon^×),\tag{5B.$\tilde{97}$}$$ where $$\varUpsilon:=\varUpsilon'+i\varUpsilon''=\pmatrix{α& β\\γ& δ}∈{\rm GL}(2\Bbb C),\quad \bar\varUpsilon:=\varUpsilon'-i\varUpsilon''=\pmatrix{\barα& \barβ\\\barγ& \barδ}\tag{5B.98}$$ is now complex. The Dirac equation (5B.48) is represented by $$\begin{multline}\hspace6em \{-\boldsymbol1⊗ iσ^2∂_1 +\boldsymbol1⊗σ^1∂_2 -m_3\boldsymbol1⊗\boldsymbol1 +m_4σ_3⊗ iσ^3\}×\\ ×[\boldsymbol1⊗\tfrac12(\varUpsilon+\bar\varUpsilon) +σ^3⊗\tfrac12(\varUpsilon-\bar\varUpsilon)]=\boldsymbol0. \hspace6em\tag{5B.99}\end{multline}$$ It spits into $$\{-iσ^2∂_1+σ^1∂_2-m_3\boldsymbol1+m_4iσ^3\}\varUpsilon=\boldsymbol0,\\ \{-iσ^2∂_1+σ^1∂_2-m_3\boldsymbol1-m_4iσ^3\}\bar\varUpsilon=\boldsymbol0.\tag{5B.100ab}$$ 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 $$E'_±=E_1±E_2=iσ_2⊗σ_±,\quad (E'^±=E^1∓E^2=iσ^2⊗σ^∓),\tag{5B.102}$$ $$\boldsymbol E'_± =σ_+⊗\varUpsilon^×σ_±\bar\varUpsilon -σ_-⊗\bar\varUpsilon^×σ_±\varUpsilon.\tag{5B.103}$$ From the immersion formula (5B.81) and the orthonormality identities (5B.59) we get the components $$\hspace-3em\eqalign{x^1±x^2 & =x^1_0±x^2_0-\tfrac12⟨∫_γ(iσ^2⊗σ^∓) [(σ_+⊗\varUpsilon^×σ_-\bar\varUpsilon -σ_-⊗\bar\varUpsilon^×σ_-\varUpsilon)\dζ'^-+\\ & \hspace5em+(σ_+⊗\varUpsilon^×σ_+\bar\varUpsilon -σ_-⊗\bar\varUpsilon^×σ_+\varUpsilon)\dζ'^+⟩\\ & =x^1_0±x^2_0+∫_γ (\tfrac18\bTr{σ^∓\varUpsilon^×σ_-\bar\varUpsilon +σ^∓\bar\varUpsilon^×σ_-\varUpsilon}\dζ'^-+\\ & \hspace5em+\tfrac18\bTr{σ^∓\varUpsilon^×σ_+\bar\varUpsilon +σ^∓\bar\varUpsilon^×σ_+\varUpsilon}\dζ'^+),}\tag{5B.104a}$$ $$\hspace-5em\eqalign{x^3 & =x^3_0-\tfrac12⟨∫_γ(iσ^2⊗{\sf s}^3) [(σ_+⊗\varUpsilon^×σ_-\bar\varUpsilon -σ_-⊗\bar\varUpsilon^×σ_-\varUpsilon)\dζ'^-+\\ & \hspace3em+(σ_+⊗\varUpsilon^×σ_+\bar\varUpsilon -σ_-⊗\bar\varUpsilon^×σ_+\varUpsilon)\dζ'^+⟩\\ & =x^3_0+∫_γ (\tfrac18\bTr{σ^3\varUpsilon^×σ_-\bar\varUpsilon +σ^3\bar\varUpsilon^×σ_-\varUpsilon)}\dζ'^-+\\ & \hspace3em+\tfrac18\bTr{σ^3\varUpsilon^×σ_+\bar\varUpsilon +σ^3\bar\varUpsilon^×σ_+\varUpsilon}\dζ'^+),}\tag{5B.104b}$$ $$\hspace-3em\eqalign{x^4 & =x^4_0-\tfrac12⟨∫_γ(σ^1⊗ i\boldsymbol1) [(σ_+⊗\varUpsilon^×σ_-\bar\varUpsilon -σ_-⊗\bar\varUpsilon^×σ_-\varUpsilon)\dζ'^-+\\ & \hspace3em+(σ_+⊗\varUpsilon^×σ_+\bar\varUpsilon -σ_-⊗\bar\varUpsilon^×σ_+\varUpsilon)\dζ'^+⟩\\ & =x^4_0-i∫_γ (\tfrac18\bTr{\varUpsilon^×σ_-\bar\varUpsilon -\bar\varUpsilon^×σ_-\varUpsilon)}\dζ'^-+\\ & \hspace3em+\tfrac18\bTr{\varUpsilon^×σ_+\bar\varUpsilon -\bar\varUpsilon^×σ_+\varUpsilon}\dζ'^+).}\tag{5B.104c}$$ With the help of the general expressions (5B.45) it gives the final formula [K96] $$x^1+x^2 =x^1_0+x^2_0+∫_γ(-β\barβ~\dζ'^-+δ\barδ~\dζ'^+),\tag{5B.105a}$$ $$x^1-x^2 =x^1_0-x^2_0+∫_γ(α\barα~\dζ'^--γ\barγ~\dζ'^+),\tag{5B.105b}$$ $$x^3+ix^4 =x^3_0+ix^4_0+∫_γ(-β\barα~\dζ'^-+δ\barγ~\dζ'^+),\tag{5B.105c}$$ $$x^3-ix^4 =x^3_0-ix^4_0+∫_γ(-α\barβ~\dζ'^-+γ\barδ~\dζ'^+).\tag{5B.105d}$$

5C. $3$-submanifolds

Only a Riemannian $3$-submanifold in $\Bbb R^4$ is worked out, since alone it goes beyond what is known. It is always in the case ii, but the potentials $v_μ$ cannot be arbitrary. By requiring that the Mainardi-Codazzi equation should have solutions for the extrinsic curvature, we get the constraint $$∂_1(v_2-v_3)=∂_2(v_3-v_1)=∂_3(v_1-v_2)=0.\tag{5C.1}$$ We set $v_1=v_2=v_3\equiv0,$ that is case i, but it is easy to adapt it to other values. The Dirac equation is $$\{E^1∂_1+E^2∂_2+E^3∂_3+mE^4\}\varPsi=\boldsymbol0,\tag{5C.2}$$ which multiplied on the left by $E_4$ is $$\{E^{\ph1}_4∂_1+E^{\ph2}_4∂_2+E^{\ph3}_4∂_3-mE\}\varPsi=\boldsymbol0.\tag{5C.3}$$ The representation and the spinor are those of (5B.49), (5B.50), and (5B.51). The Dirac equation is represented as $$\begin{multline}\hspace6em \{-iσ^3⊗(σ^1∂_1+σ^2∂_2 +σ^3∂_3)-m\boldsymbol1⊗\boldsymbol1\}×\\ ×[\boldsymbol1⊗\tfrac12(\varPhi_1+\varPhi_2) +σ_3⊗\tfrac12(\varPhi_1-\varPhi_2)]=\boldsymbol0, \hspace6em\tag{5C.4}\end{multline}$$ and splits into $$\eqalign{ \{iσ^1∂_1+iσ^2∂_2+iσ^3∂_3+m\boldsymbol1\}\varPhi_1& =0,\\ \{iσ^1∂_1+iσ^2∂_2+iσ^3∂_3-m\boldsymbol1\}\varPhi_2& =0.}\tag{5C.5}$$ It splits further to $$\left\{\eqalign{ 2∂χ_1+∂_3φ_1=imφ_1,\quad& 2∂χ_2+∂_3φ_2=-imφ_2,\\ 2\bar∂φ_1+∂_3χ_1=imχ_1,\quad& 2\bar∂φ_2+∂_3χ_2=-imχ_2. }\right.\tag{5C.6}$$ All the additional constraints (3A.38) are equivalent to the following equations and their complex conjugate : $$\displaylines{ -i(z^1_{2/1}-iz^1_{2/2})=z^3_{1/3}-iz^3_{2/3},\\ -2iz^1_{2/3}=0,\\ \tfrac12[(z^3_{1/1}-iz^3_{1/2})+i(z^2_{3/1}-iz^2_{3/2})]=0,\\ -i(z^2_{3/1}+z^3_{1/2})=0,}\tag{5C.7}$$ which in terms of the spinor components read $$\displaylines{ \sum_{n=1,2}\barφ_n∂φ_n-φ_n∂\barφ_n +χ_n∂\barχ_n-\barχ_n∂χ_n =\sum_{n=1,2}φ_n∂_3\barχ_n-\barχ_n∂_3φ_n,\\ \sum_{n=1,2}\barφ_n∂_3φ_n-φ_n∂_3\barφ_n +χ_n∂_3\barχ_n-\barχ_n∂_3χ_n=0,\\ \sum_{n=1,2}φ_n∂\barχ_n-\barχ_n∂φ_n=0,\\ \sum_{n=1,2}\barφ_n∂χ_n-χ_n∂\barφ_n +\barχ_n\bar∂φ_n-φ_n\bar∂\barχ_n=0.}\tag{5C.8}$$ The differential forms (3A.33) are $ω^α=\rho^{-1/2}\dξ^α.$ the immersion formula is then $$\boldsymbol X=\boldsymbol X_0 +∫_γ\rho^{-1/2}(\boldsymbol E_1\dξ^1+\boldsymbol E_2\dξ^2+\boldsymbol E_3\dξ^3),\tag{5C.9}$$ where $$\rho=\tfrac12(φ_1\barφ_1+χ_1\barχ_1+φ_2\barφ_2+χ_2\barχ_2).\tag{5C.10}$$

The step numbers and the tangent vectors are $$E_±=E_1±iE_2=σ_2⊗iσ_\pm,\quad E_3=σ_2⊗iσ_3.\tag{5C.11}$$ $$\eqalign{\boldsymbol E_± & =σ_+⊗\tfrac12\varPhi_1^†σ_±\varPhi_2 -σ_-⊗\tfrac12\varPhi_2^†σ_±\varPhi_1,\\ \boldsymbol E_3 & =σ_+⊗\tfrac12\varPhi_1^†σ_3\varPhi_2 -σ_-⊗\tfrac12\varPhi_2^†σ_3\varPhi_1.}\tag{5C.12}$$ With the orthonormality identities (5B.59) and the immersion formula $$\eqalign{ \boldsymbol X & =\tfrac12(x^1+ix^2)E_-+\tfrac12(x^1-ix^2)E_++x^3E_3+x^4E_4\\ & =\boldsymbol X_0+\tfrac12∫_γ\rho^{-1/2}(2\boldsymbol E_3\dξ'^3 +\boldsymbol E_-\dζ'+\boldsymbol E_+\d\barζ'),}\tag{5C.13}$$ we get the components $$\hspace-5em\eqalign{x^1±ix^2 & =x^1_0±ix^2_0 -\tfrac12⟨∫_γ\rho^{-1/2}(σ^2⊗ iσ^±)\times\\ & \hspace4em\times[2(σ_+⊗\tfrac12\varPhi_1^†σ_3\varPhi_2 -σ_-⊗\tfrac12\varPhi_2^†σ_3\varPhi_1)\dξ'^3+\\ & \hspace6em+(σ_+⊗\tfrac12\varPhi_1^†σ_-\varPhi_2-σ_ -⊗\tfrac12\varPhi_2^†σ_-\varPhi_1)\dζ'+\\ & \hspace6em+(σ_+⊗\tfrac12\varPhi_1^†σ_+\varPhi_2-σ_ -⊗\tfrac12\varPhi_2^†σ_+\varPhi_1)\d\barζ']⟩\\ & =x^1_0±ix^2_0 +∫_γ\rho^{-1/2}(\tfrac14\bTr{σ^±\varPhi_1^†σ_3\varPhi_2 +σ^±\varPhi_2^†σ_3\varPhi_1}\dξ'^3+\\ & \hspace5em+\tfrac18\bTr{σ^±\varPhi_1^†σ_-\varPhi_2 +σ^±\varPhi_2^†σ_-\varPhi_1}\dζ'+\\ & \hspace5em+\tfrac18\bTr{σ^±\varPhi_1^†σ_+\varPhi_2 +σ^±\varPhi_2^†σ_+\varPhi_1}\d\barζ'),}\tag{5C.14a}$$ $$\hspace-2em\eqalign{x^3 & =x^3_0-\tfrac12⟨∫_γ\rho^{-1/2}(σ^2⊗ iσ^3)[ 2(σ_+⊗\tfrac12\varPhi_1^†σ_3\varPhi_2 -σ_-⊗\tfrac12\varPhi_2^†σ_3\varPhi_1)\dξ'^3+\\ & \hspace3em+(σ_+⊗\tfrac12\varPhi_1^†σ_-\varPhi_2 -σ_-⊗\tfrac12\varPhi_2^†σ_-\varPhi_1)\dζ'+\\ & \hspace3em+(σ_+⊗\tfrac12\varPhi_1^†σ_+\varPhi_2 -σ_-⊗\tfrac12\varPhi_2^†σ_+\varPhi_1)\d\barζ']⟩\\ & =x^3_0+∫_γ\rho^{-1/2}(\tfrac14\bTr{σ^3\varPhi_1^†σ_3\varPhi_2 +σ^3\varPhi_2^†σ_3\varPhi_1}\dξ'^3+\\ & \hspace3em+\tfrac18\bTr{σ^3\varPhi_1^†σ_-\varPhi_2 +σ^3\varPhi_2^†σ_-\varPhi_1}\dζ'+\\ & \hspace5em+\tfrac18\bTr{σ^3\varPhi_1^†σ_+\varPhi_2 +σ^3\varPhi_2^†σ_+\varPhi_1}\d\barζ'),}\tag{5C.14b}$$ $$\hspace-2em\eqalign{x^4 & =x^4_0-\tfrac12⟨∫_γ\rho^{-1/2}(σ^1⊗ i\boldsymbol1) [2(σ_+⊗\tfrac12\varPhi_1^†σ_3\varPhi_2 -σ_-⊗\tfrac12\varPhi_2^†σ_3\varPhi_1)\dξ'^3+\\ & \hspace3em+(σ_+⊗\tfrac12\varPhi_1^†σ_+\varPhi_2 -σ_-⊗\tfrac12\varPhi_2^†σ_+\varPhi_1)\dζ'+\\ & \hspace3em+(σ_+⊗\tfrac12\varPhi_1^†σ_-\varPhi_2 -σ_-⊗\tfrac12\varPhi_2^†σ_-\varPhi_1)\d\barζ']⟩\\ & =x^4_0-i∫_γ\rho^{-1/2}(\tfrac14\bTr{\varPhi_1^†σ_3\varPhi_2 -\varPhi_2^†σ_3\varPhi_1}\dξ'^3+\\ & \hspace3em+\tfrac18\bTr{\varPhi_1^†σ_+\varPhi_2 -\varPhi_2^†σ_+\varPhi_1}\dζ'+\\ & \hspace5em+\tfrac18\bTr{\varPhi_1^†σ_-\varPhi_2 -\varPhi_2^†σ_-\varPhi_1}\d\barζ').}\tag{5C.14c}$$ With the help of the general expressions (5B.15) and $$\bTr{σ^+\varPhi^†_mσ_3\varPhi_n} =-2φ_mχ_n-2φ_nχ_m,\tag{5C.15a}$$ $$\bTr{σ^-\varPhi^†_mσ_3\varPhi_n} =-2\barφ_m\barχ_n-2\barφ_n\barχ_m,\tag{5C.15b}$$ $$\hspace-1em\bTr{σ^3\varPhi^†_mσ_3\varPhi_n} =φ_n\barφ_m+φ_m\barφ_n-χ_n\barχ_m-χ_m\barχ_n,\tag{5C.15c}$$ $$\bTr{\varPhi^†_mσ_3\varPhi_n} =φ_n\barφ_m-φ_m\barφ_n -χ_n\barχ_m+χ_m\barχ_n,\tag{5C.15d}$$ it gives the final formula $$\hspace-2emx^1+ix^2=x^1_0+ix^2_0+∫_γ\rho^{-1/2}[-(φ_1χ_2+φ_2χ_1)\dξ'^3 +φ_1φ_2\,\dζ'-χ_1χ_2\,\d\barζ'],\tag{5C.16a}$$ $$\hspace-2emx^1-ix^2=x^1_0-ix^2_0+∫_γ\rho^{-1/2} [-(\barφ_1\barχ_2+\barφ_2\barχ_1)\dξ'^3 -\barχ_1\barχ_2\,\dζ'+\barφ_1\barφ_2\,\d\barζ'],\tag{5C.16b}$$ $$\hspace-2emx^3+ix^4=x^3_0+ix^4_0+∫_γ\rho^{-1/2} [\phantom-(φ_2\barφ_1-χ_2\barχ_1)\dξ'^3 +\barχ_1φ_2\,\dζ'+\barφ_1χ_2\,\d\barζ'],\tag{5C.16c}$$ $$\hspace-2emx^3-ix^4=x^3_0-ix^4_0+∫_γ\rho^{-1/2} [\phantom-(φ_1\barφ_2-χ_1\barχ_2)\dξ'^3 +φ_1\barχ_2\,\dζ'+χ_1\barφ_2\,\d\barζ'].\tag{5C.16d}$$

Conclusion

The main result presented here is the demonstration that a submanifold immersed in a flat space of higher dimension can be constructed from a solution of a Dirac-like equation, provided that it belongs to the special Clifford group. This is true for arbitrary dimension, codimension and signature, while it was known for a surface only.

The Dirac equation allows to prescribe the Gauß map, and then to lift it to an element of the special Clifford group solution of the equation. But for any case apart from a surface, this gives many transcendant equations that are not solved yet. The additional constraints are not easily handled too. This is a big challenge to further develop the theory since it has important applications.

The main application of this framework is the inverse scattering transform, a method used to solve nonlinear partial differential equations. 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 some submanifold (4C.14), in general it is a nonlinear partial differential equation of third order that can be simplified in special case and with additional data. When it can be solved, the function $u$ can then be determined from (4C.18), and it is the solution of the nonlinear equation. The lift of the Gauß map is an orthogonal 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, this equation the considered nonlinear partial differential equation is the Gauß equation (2B.GR), the Weingarten equation (2B.GW) 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, while the compatibility condition of the Lax set is the zero-curvature condition assuring that a $\varPsi$ function exists. 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. Moreover the Lax set is no longer equivalent to a Dirac equation.

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 in fact 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 that 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.