Spinor Formulation of the Differential Geometry of Curves

Home , Differentiable curve

Case 1 Revista Colombiana de Matem a-acute ti cas Case 2 Volumen 38 open parenthesis 2004 closing parenthesis\ begin { a l comma i g n ∗} p ginas 27 endash 34 \ l eSpinor f t . \ acute .. formulation{a}\ begin .. of{ thea l i g n e d } & Revista Colombiana de Matem \acute{a} t idifferential cas \\ .. geometry .. of .. curves &Gerardo Volumen Francisco 38 .. Torres ( del 2004 Castillo ) , p ginas 27 −− 34 \end{ a l i g n e d }\ right . \endGuadalupe{ a l i g n ∗} S a-acute nchez Barrales Universidad Aut acute-o noma de Puebla comma M e-acute xico \ centerlinex41 x62 x73{ x74Spinor x72 x61\quad x63 x74formulation x2e .. It is shown\quad thato f the the Frenet} equations for curves in x52 to the power of 3 can be RevistaColombianadeMatema´ticas \ centerline { differential \quada´ geometry \quad o f \quad curves } reduced to a single equation for a vectorVolumen38(2004) with two complex, pginas27 components− −34 and some examples of the usefulness of this representation are given period \ centerlineKeywords and{Gerardo phrases period Francisco Frenet equations\quad Torres comma spinors del Castillo period } 2000 Mathematics Subject ClassificationSpinor period formulation Primary : 55 A 4 period Secondary of the : 1 5 A 66 period \ centerline1 period .. Introduction{Guadalupedifferential S $ \acute{a} geometry$ nchez Barrales of} curves In differential geometry theGerardo curves and surfacesFrancisco in R to the Torres power of 3 del are usually Castillo studied em hyphen \ centerlineploying the{ vectorUniversidad formalism periodAutGuadalupe $In\ theacute case{ ofo} aS differentiable$a´ nomanchez de Barralescurve Puebla comma ,M at each $ \ pointacute{e} $ x i ca o triad} of mutually orthogonalUniversidad unit vectors openAut parenthesiso´ noma de called Puebla tangent , M commae´ xico normal and binor hyphen x41x62x73x74x72x61x63x74x2e It is shown that the Frenet equations for curves in x523 can be \ centerlinemal closing{ parenthesis$x41reduced is x62 constructed to a x73 single and equation x74 the rates for x72 ofa vector change x61 with ofthese two x63 complex vectors x74 components along x2e$ the curve and\quad It is shown that the Frenet equations for curves in $ x52define ˆ{ the3 curvature}$ can and be torsion}some examples of the curve of the period usefulness These of two this functions representation characterize are given . the curve completely except for itsKeywords position and and phrases orientation . Frenet in spaceequations period , spinors In a sim. hyphen \ centerline { reduced to a single equation for a vector with two complex components and } i lar manner comma2000 at Mathematics each point Subjectof a smooth Classification surface . aPrimary triad of mutually: 55 A 4 . orthogonal Secondary : 1 5 A 66 . unit vectors can be defined in such a way that1 one . ofIntroduction these vectors is normal to \ centerline {some examples of the usefulness of this representation are given . } the surface periodIn Then differential the rate geometry of variation the of curves the normal and surfaces unit vector in R to3 are the surfaceusually studied em - along the directionsploying of the the other vector two formalism vectors determines . In the case the of curvature a differentiable of the curve , at each point \ centerlinesurface period{Keywordsa triad of and mutually phrases orthogonal . Frenet unit vectors equations ( called , tangent spinors , normal . } and binor - Given the importancemal ) of is the constructed triads of andmutually the rates orthogonal of change unit of vectors these vectors in along the curve \ centerlinedifferential geometry{2000define Mathematics comma the curvature it is of interestSubject and torsion that Classification each of the such curve triad . These can . be Primary two expressed functions : in 55 characterize A 4 . Secondary : 1 5 A 66 . } terms of a singlethe vector curve with completely two complex except components for its position comma and called orientation a sp in or in openspace square . In a bracket sim - 1 endash\ centerline 3 closing{1 square . i\ larquad bracket mannerIntroduction period , at each point} of a smooth surface a triad of mutually orthogonal The aim of this paperunit vectors is to show can that be defined the basic in such equations a way of that the one differential of these vectors is normal to the \noindentgeometry ofIn curves differentialsurface open . parenthesis Then geometry the rate the of Frenet variation the equationscurves of the and normal closing surfaces unit parenthesis vector in to can the $ be R surface êxpressed{ 3 }$ in are a usually studied em − compactploying and the vectoralong the formalism directions of . the In other the two case vectors of determinesa differentiable the curvature curve of the surface , at each point 2 7 . \noindent a triad of mutuallyGiven the importance orthogonal of the unit triads vectors of mutually ( orthogonal called tangent unit vectors , normal in and binor − differential geometry , it is of interest that each such triad can be expressed in \noindent mal )terms is constructed of a single vector and with the two complexrates of components change , of called these a sp in vectors or [ 1 – along3 ] . the curve The aim of this paper is to show that the basic equations of the differential \noindent definegeometry the curvature of curves ( the and Frenet torsion equations of )the can curve be expressed . These in a compact two functions and characterize 2 7 \noindent the curve completely except for its position and orientation in space . In a sim − i lar manner , at each point of a smooth surface a triad of mutually orthogonal unit vectors can be defined in such a way that one of these vectors is normal to the surface . Then the rate of variation of the normal unit vector to the surface

\noindent along the directions of the other two vectors determines the curvature of the s u r f a c e .

\ hspace ∗{\ f i l l }Given the importance of the triads of mutually orthogonal unit vectors in

\noindent differential geometry , it is of interest that each such triad can be expressed in

\noindent terms of a single vector with two complex components , called a sp in or [ 1 −− 3 ] .

\noindent The aim of this paper is to show that the basic equations of the differential

\noindent geometry of curves ( the Frenet equations ) can be expressed in a compact and

\ centerline {2 7 } 2 8 .. G period F period TORRES DEL CASTILLO ampersand G period S A-acute sub NCHEZ BARRALES\noindent 2 8 \quad G . F . TORRES DEL CASTILLO $ \& $ G . S $ \acute{A} { NCHEZ }$ BARRALESuseful way making use of spinors period In Section 2 the basic elements about spinors are briefly presented open parenthesis a more complete elementary treatment can be found in \noindentopen squareuseful bracket way 1 comma making 3 closing use of square spinors bracket . closing In Section parenthesis 2 thesemicolon basic in Sectionelements 3 the about spinors spinorare equivalent briefly of presented the Frenet equations ( a more for complete a curve is elementary treatment can be found in obtained and in Sec period 4 some examples of it s application are given period 2 8 G . F . TORRES DEL CASTILLO & G.S A´ BARRALES \noindent2 period ..[ Orthonormal 1 , 3 ] ) bases ; in and Section spinors 3 the spinorNCHEZ equivalent of the Frenet equations for a curve is useful way making use of spinors . In Section 2 the basic elements about spinors obtainedThe group and of rotations in Sec about . 4 the some origin examples in R to the of power it s of application 3 open parenthesis are denoted given .as SO open are briefly presented ( a more complete elementary treatment can be found in parenthesis 3 closing parenthesis closing parenthesis is known [ 1 , 3 ] ) ; in Section 3 the spinor equivalent of the Frenet equations for a curve is \ centerlineto be homomorphic{2 . \quad to theOrthonormal group of unitary bases complex and 2 times spinors 2 matrices} with unit obtained and in Sec . 4 some examples of it s application are given . determinant open parenthesis denoted as SU open parenthesis 2 closing parenthesis closing parenthesis 2 . Orthonormal bases and spinors period\noindent In factThe comma group there of is a rotations two hyphen to about hyphen the one origin homomorphism in $ R ˆ{ 3 } ( $ denoted as SO ( 3 ) ) is known The group of rotations about the origin in 3 ( denoted as SO ( 3 ) ) is known to be toof be SU homomorphic open parenthesis to 2 theclosing group parenthesis of unitary onto SO complex openR parenthesis $ 2 3\ closingtimes parenthesis2 $ matrices period with unit homomorphic to the group of unitary complex 2×2 matrices with unit determinant Whereasdeterminant the elements ( denoted of SO open as parenthesisSU ( 2 ) 3 ) closing . In parenthesis fact , there act on is points a two of R− toto the− powerone ofhomomorphism 3 ( denoted as SU ( 2 ) ) . In fact , there is a two - to - one homomorphism open parenthesis that of SU ( 2 ) onto SO ( 3 ) . Whereas the elements of SO ( 3 ) act on points of \noindentis comma vectorsof SU with ( 2 three ) onto real components SO ( 3 ) closing . Whereas parenthesis the comma elements the elements of SO of ( SU 3 )open act paren- on points of 3( that is , vectors with three real components ) , the elements of SU ( 2 ) act thesis$ R ˆ 2{ closing3 } parenthesis( $R that act on vectors on vectors with two complex components which are called spinors ( see also , for iswith , vectors two complex with components three real which components are called spinors ) , open the parenthesis elements see of also SU comma ( 2 ) for act example on vectors example , commawith two complex components which are called spinors ( see also , for example , [ 4 , 5 ] ) . open square bracket 4 comma 5 closing square bracket closing parenthesis period An explicit way of exhibiting this homomorphism consists in noticing that \noindentAn explicit[ way 4 ,of 5 exhibiting ] ) . this homomorphism consists in noticing that each spinor each spinor \ hspaceEquation:∗{\ openf i l l } parenthesisAn explicit 1 closing way parenthesis of exhibiting .. psi = Row this 1 psi homomorphism 1 Row 2 psi 2 . consists in noticing that defines three vectors a comma b comma c in R to the power ψ1 of 3 by means of \noindent each s p i n o r ψ = (1) a plus i b = psi to the power of t sigma psi comma .. c = minusψ2 psi-hatwide to the power of t sigma psi comma open parenthesis 2 closing parenthesis \ begin { a l i g n ∗} where sigma is a vectordefines whose three vectors Cartesiana components, b , c ∈ are3 by the means complex of symmetric \ psi = \ l e f t (\ begin { array }{ c}\ psi R1 \\\ psi 2 \end{ array }\ right )\ tag ∗{$ ( 2 times 2 matrices a +i b = ψtσψ, c = −ψbtσψ, (2) where σ is a vector whose Cartesian 1 ) $} Equation: open parenthesiscomponents 3 areclosing the parenthesis complex symmetric .. sigma sub 1 = Row 1 1 0 Row 2 0 minus 1 . comma \end{ a l i g n ∗} sigma sub 2 = Row 12 i× 02 Row matrices 2 0 i . comma sigma sub 3 = Row 1 0 minus 1 Row 2 minus 1 0 . comma the superscript t denotes transposition and psi-hatwide is the mate open square bracket 3 closing square bracket\noindent open parenthesisdefines three or conjugate vectors open asquare , b bracket , c $ 1 closing\ in squareR ˆ{ bracket3 }$ closing by means parenthesis o f 1 0 i 0 0 −1 of psi comma σ1 = , σ2 = , σ3 = , (3) aEquation: $ + i open $ b parenthesis $ = \ 4psi closingˆ0{ parenthesist−1}\sigma .. psi-hatwide0\ psii equiv, minus $ \quad− Row1 1 0c 0 1 $ Row = 2− minus \ 1widehat{\ psi } ˆ{ t } 0\sigma . psi = minus\ psi Row 1, 0 1 Row ( 2 2 minus ) 1 $ 0 . Row 1 psi 1 Row 2 psi 2 . = Row 1 minus psi 2 Row 2 psi 1 .where comma $ \sigmathe$ superscript is a vector t denotes whose transposition Cartesian and ψb componentsis the mate [ 3 are ] ( or the conjugate complex [ 1 ] symmetric where the bar denotes) of ψ, complex conjugation period Thus comma the vectors a comma b comma and c are\noindent $ 2 \times 2 $ matrices explicitly given by 0 1 0 1 ψ1 −ψ2 \ begin { a l i g n ∗} ψ ≡ − ψ = − = , (4) Line 1 a plus i b = open parenthesisb psi−1 sub 0 1 to the power−1 of 2 0 minusψ psi2 sub 2 toψ the1 power of 2 comma i open\sigma parenthesis{ 1 } psi= sub\ 1l eto f t the(\ powerbegin of{ array 2 plus}{ psicc sub} 1 2 to & the 0 power\\ of0 2 & closing− parenthesis1 \end{ array comma}\ right ) , \sigma { 2 } = \ l e f t (\ begin { array }{ cc } i & 0 \\ 0 & i \end{ array }\ right ) minus 2 psi 1 psi 2 closingwhere parenthesis the bar denotes comma complex Line 2 conjugation c = open parenthesis . Thus , the psi vectors 2 psi 1 plusa , b psi, 1 and psic 2 commaare , \sigma { 3 } = \ l e f t (\ begin { array }{ cc } 0 & − 1 \\ − 1 & 0 \end{ array }\ right ) i psi 2 psi 1 minus i psiexplicitly 1 psi 2 given comma by bar psi 1 bar to the power of 2 minus bar psi 2 bar to the power of 2, closing\ tag ∗{ parenthesis$ ( 3 ) $} \end{ a l i g n ∗} 2 2 2 2 and by means of an explicit computation onea + i findsb = that ( aψ1 comma− ψ2, i b(ψ comma1 + ψ2) and, −2ψ c1 areψ2) mutually, orthogonal and bar a bar = barc b= bar (ψ =2ψ bar1 + cψ bar1ψ2 =, iψ psi2ψ to1 − theiψ1 powerψ2, | ψ of1 |2 t− psi | ψ period2 |2) Furthermore a times\noindent b timesthe c greater superscript 0 period t denotes transposition and $ \widehat{\ psi } $ is the mate [ 3 ] ( or conjugate [ 1 ] ) o fIt may $ \ bepsi pointed,and $out by that means psi-hatwide of an explicit transforms computation under the one SU finds open that parenthesisa , b , and 2c closingare mutually parenthesis t transformations orthogonal and | a |=| b |=| c |= ψ ψ. Furthermore a × b · c > 0. \ beginin exactly{ a l i g the n ∗} same wayIt as may psi be does pointed comma out that that isψ commab transforms U-hatwidest under the psi SU = U ( hatwide-psi 2 ) transformations sub comma \widehat{\ psi }\equiv − \ l e f t (\ begin { array }{ cc } 0 & 1 \\ − 1 & 0 \end{ array }\ right ) for any U in SU openin parenthesis exactly the 2 same closing way parenthesis as ψ does period , that is , Uψb = Uψb, for any U ∈ SU ( 2 ) . \ psiConversely= − comma \ l e given f tConversely(\ threebegin mutually{ ,array given three}{ orthogonalcc } mutually0 vectors & orthogonal 1 \\ of the − same vectors1 magnitude & of the 0 \ sameend comma{ magnitudearray }\ right , ) \ l e f t (\ begin { array }{ c}\ psi 1 \\\ psi 2 \end{ array }\3right ) = \ l e f t (\ begin { array }{ c} − \ psi 2 \\ a comma b commaa c, inb R, c to∈ theR power, such of that 3 commaa × b such· c that> 0, athere times exists b times a spinor c greater , defined 0 comma up to there exists\ psi a spinor1 \end comma{ arraysign defined ,}\ suchright up that to), sign( 2 ) comma holds\ tag .∗{$ ( 4 ) $} \endsuch{ a thatl i g n open∗} parenthesis 2 closing parenthesis holds period \noindent where the bar denotes complex conjugation . Thus , the vectors a , b , and c are explicitly given by

\ [ \ begin { a l i g n e d } a + i b = ( \ psi ˆ{ 2 } { 1 } − \ psi ˆ{ 2 } { 2 } , i ( \ psi ˆ{ 2 } { 1 } + \ psi ˆ{ 2 } { 2 } ), − 2 \ psi 1 \ psi 2 ) , \\ c = ( \ psi 2 \ psi 1 + \ psi 1 \ psi 2 , i \ psi 2 \ psi 1 − i \ psi 1 \ psi 2 , \mid \ psi 1 \mid ˆ{ 2 } − \mid \ psi 2 \mid ˆ{ 2 } ) \end{ a l i g n e d }\ ]

\noindent and by means of an explicit computation one finds that a , b , and c are mutually

\noindent orthogonal and $ \mid $ a $ \mid = \mid $ b $ \mid = \mid $ c $ \mid = \ psi ˆ{ t }\ psi . $ Furthermore a $ \times $ b $ \cdot $ c $ > 0 . $

\ hspace ∗{\ f i l l } It may be pointed out that $ \widehat{\ psi } $ transforms under the SU ( 2 ) transformations

\noindent in exactly the same way as $ \ psi $ does , that is $ , \widehat{U} \ psi = U \widehat{\ psi } { , }$ f o r any $ U \ in $ SU ( 2 ) .

\ hspace ∗{\ f i l l } Conversely , given three mutually orthogonal vectors of the same magnitude ,

\noindent a , b , c $ \ in R ˆ{ 3 } , $ such that a $ \times $ b $ \cdot $ c $ > 0 , $ there exists a spinor , defined up to sign , such that ( 2 ) holds . S PINOR FORMULATION OF THE DIFFERENTIAL GEOMETRY OF CURVES 2 9 t For any U ∈ SU ( 2 ) , the spinor ψ0 = Uψ satisfies ψ0 ψ0 = ψtψ; therefore , the magnitudes of the three vectors a0, b0, c0 defined by ψ0 are equal to those of the three vectors a , b , c corresponding to ψ. Hence , each element of SU ( 2 ) induces a transformation that sends the right - handed orthogonal basis { a , b , c } of R3 into the right - handed orthogonal basis {a0, b0, c0}, that is , an element of SO ( 3 ) . This correspondence between elements of SU ( 2 ) and elements of SO ( 3 ) is two to one since U and −U yield the same element of SO ( 3 ) . Making use of the foregoing definitions it can be shown that if φ and ψ are two arbitrary spinors

t t φ σψ = −φb σψb, (5) and for any pair of complex numbers , a, b,

(aφ + bψ)b= aφb + bψb. (6) Furthermore ,

ψbb = −ψ. (7)

Making use of these properties it can be readily seen that the vector c , given by ( 2 ) , is real . The correspondence between spinors and orthogonal bases given by ( 2 ) is two to one ; the spinors ψ and −ψ correspond to the same ordered orthonormal basis { a , b , c } , with | a |=| b |=| c | and a × b · c > 0. It is important to notice that the ordered triads { a , b , c } , { b , c , a } and { c , a , b } correspond to different spinors . The symmetry of the matrices ( 3 ) amounts to φtσψ = ψtσφ for any pair of 0 1 spinors φ and ψ. ( The matrices ( 3 ) are the products of the matrix −1 0 by the Pauli matrices employed in physics [ 2 , 3 ] . ) With the conventions chosen in ( 2 ) – ( 4 ) , taking ψ = (1, 0)t one finds that ψb = (0, 1)t and the triad { a , b , c } is the canonical basis of R3. If ψ is a spinor different from zero , the set {ψ, ψb} is linearly independent ( using the complex numbers as scalars ) , which follows from the fact that the determinant of the matrix formed by the components of ψ and ψb is    ψ1 −ψ2  2 2   =| ψ1 | + | ψ2 |  ψ2 ψ1  and this sum is equal to zero if and only if ψ1 and ψ2 are simultaneously equal to zero . 3. F − renet equations S PINOR FORMULATION OF THE DIFFERENTIAL GEOMETRY OF CURVES .. 2 9 \ hspaceFor any∗{\ Uf in i l SU l }S open PINOR parenthesis FORMULATION 2 closing OFparenthesis THE DIFFERENTIAL comma the spinor GEOMETRY psi to the OF power CURVES of prime\quad = 2 9 U psi satisfies psi to the power of prime to the power of t psi to the power of prime = psi to the power of t psiFor semicolon any $ therefore U \ in comma$ SU( 2 ) , the spinor $ \ psi ˆ{\prime } = U \ psi $ s athe t i s f magnitudes i e s $ \ psi of theˆ{\ threeprime vectorsˆ{ a tot the}}\ powerpsi ofˆ prime{\prime comma} b to= the\ powerpsi ˆ of{ primet }\ commapsi c to; the$ power therefore of prime , defined by psi to the power of prime are equal to those theof the magnitudes three vectors of a comma the three b comma vectors c corresponding $ a ˆ{\ to psiprime period} Hence, comma b ˆ{\ eachprime element} of, SU For any curve α : I → 3 such that dα(t)/dt 6= 0, there exists a function openc ˆ{\ parenthesisprime } 2$ closing defined parenthesis by $ \ psi ˆR{\prime }$ are equal to those s = s(t), called the arclength , such that if the curve is parametrized by s, ofinduces the three a transformation vectors that a , sends b , the c corresponding right hyphen handed to orthogonal $ \ psi basis. open$ Hence brace a , comma each b element of SU ( 2 ) | dα/ds |= 1. Then ,T ≡ dα/ds is a unit vector , called the tangent of α and if commainduces c closing a transformation brace that sends the right − handed orthogonal basis \{ a , b , c \} dT/ds 6= 0, the curvature of α, κ, is defined by κ ≡| dT/ds |; thus , dT/ds = κN, of R to the power of 3 into the right hyphen handed orthogonal basis open brace a to the power of prime where N is some unit vector , called normal of α, and T and N are orthogonal to comma\noindent b to theo f power $ R of ˆ{ prime3 }$ comma into c to the the right power− of primehanded closing orthogonal brace comma basis that is $ comma\{ ana ˆ{\prime } element, b ôf{\prime } , c ˆ{\prime }\} , $ that is , an element of SOSO ( open 3 ) parenthesis . This correspondence 3 closing parenthesis between period This elements correspondence of SU ( between 2 ) and elements elements of SU ofopen SO ( 3 ) parenthesisis two to 2 closing one since parenthesis $U$ and elements and $ of− SO openU $ parenthesis yield the 3 closing same parenthesis element of SO ( 3 ) . is two to one since U and minus U yield the same element of SO open parenthesis 3 closing parenthesis period\ hspace ∗{\ f i l l }Making use of the foregoing definitions it can be shown that if $ \Makingphi $ use and of the $ foregoing\ psi $ definitions are it can be shown that if phi and psi are two arbitrary spinors \noindentEquation:two open arbitraryparenthesis 5 spinorsclosing parenthesis .. phi to the power of t sigma psi = minus phi-hatwide to the power of t sigma hatwide-psi sub comma \ beginand for{ a anyl i g n pair∗} of complex numbers comma a comma b comma \phiEquation:ˆ{ t open}\ parenthesissigma 6\ psi closing parenthesis= − \ ..widehat open parenthesis{\phi} aˆ phi{ t plus}\ b psisigma closing parenthesis\widehat{\ psi } { , }\ tag ∗{$ ( hatwide5 ) $=} a phi-hatwide plus b hatwide-psi sub period \endFurthermore{ a l i g n ∗} comma Equation: open parenthesis 7 closing parenthesis .. psi-hatwide-hatwide = minus psi period \noindentMaking useand of these for properties any pair it canof complexbe readily seen numbers that the $ vector , ca comma , given b , $ by open parenthesis 2 closing parenthesis comma is real period \ beginThe correspondence{ a l i g n ∗} between spinors and orthogonal bases given by open parenthesis 2 closing parenthesis is( a \phi + b \ psi ) \widehat{} = a \widehat{\phi} + b \widehattwo to one{\ psi semicolon} { . the}\ spinorstag ∗{ psi$ ( and minus 6 ) psi $} correspond to the same ordered orthonormal \endbasis{ a l open i g n ∗} brace a comma b comma c closing brace comma with bar a bar = bar b bar = bar c bar and a times b times c greater 0 period It is important to notice \noindentthat the orderedFurthermore triads open , brace a comma b comma c closing brace comma open brace b comma c comma a closing brace and open brace c comma a comma b closing brace correspond to different \ beginspinors{ a periodl i g n ∗} \widehatThe symmetry{\widehat of the{\ matricespsi }} open= parenthesis− \ psi 3 closing. parenthesis\ tag ∗{$ ( amounts 7 to ) phi $} to the power of t sigma\end{ psia l i = g n psi∗} to the power of t sigma phi for any pair of spinors phi and psi period open parenthesis The matrices open parenthesis 3 closing parenthesis are the products\noindent of theMaking matrix Rowuse 1 of 0 1 these Row 2 minusproperties 1 0 . it can be readily seen that the vector c , given by(2)by the Pauli , matrices is real employed . in physics open square bracket 2 comma 3 closing square bracket period closing parenthesis With the conventions chosen Thein correspondence open parenthesis 2 closing between parenthesis spinors endash and open orthogonal parenthesis bases 4 closing given parenthesis by ( 2 comma ) is taking psitwo = open to one parenthesis ; the 1 spinors comma 0 closing $ \ psi parenthesis$ and to the$ − power \ psi of t one$ finds correspond that psi-hatwide to the = same open ordered orthonormal parenthesisb a s i s \{ 0 commaa , b 1 , closing c \} parenthesis, with $ to\ themid power$ ofa t $and\mid the triad= open\mid brace$ a comma b $ b\mid comma= c closing\mid $ brace c $ \mid $ and a $ \times $ b $ \cdot $ c $ > 0 . $ It is important to notice thatis the the canonical ordered basis triads of R to the\{ powera , ofb 3 , period c \} , \{ b , c , a \} and \{ c , a , b \} correspond to different If psi is a spinor different from zero comma the set open brace psi comma psi-hatwide closing brace is linearly\noindent independents p i n o r s . open parenthesis using the complex numbers as scalars closing parenthesis comma which follows from the\ hspace fact that∗{\ thef i l l }The symmetry of the matrices ( 3 ) amounts to $ \phi ˆ{ t }\sigma \ psideterminant= \ ofpsi theˆ matrix{ t }\ formedsigma by the components\phi $ for of psi any and pair psi-hatwide of is Row 1 psi 1 minus psi 2 Row 2 psi 2 psi 1 . = bar psi 1 bar to the power of 2 plus bar psi 2 bar to the power\noindent of 2 s p i n o r s $ \phi $ and $ \ psi . ( $ The matrices ( 3 ) are the products of the matrix $\ land e f t this(\ begin sum is{ array equal to}{ zerocc } if0 and & only 1 if\\ psi 1− and1 psi &2 are 0 simultaneously\end{ array }\ equalright )$ to zero period \noindent3 period F-rby sub the enet Pauli equations matrices employed in physics [ 2 , 3 ] . ) With the conventions chosen inFor ( any 2 ) curve−− ( alpha 4 ) : ,I right taking arrow $ R\ topsi the power= of ( 3 such 1 that , d alpha0 ) open ˆ{ parenthesist }$ one t closing finds that parenthesis$ \widehat slash{\ psi dt equal-negationslash} = ( 0 0 comma , 1 .. there ) ˆ{ existst }$ a function and the triad \{ a , b , c \} s = s open parenthesis t closing parenthesis comma .. called the arclength comma .. such that if the curve\noindent is parametrizedis the by canonical s comma basis of $ R ˆ{ 3 } . $ bar d alpha slash ds bar = 1 period .. Then comma T equiv d alpha slash ds is a unit vector comma calledI f $ the\ psi tangent$ of is alpha a spinor and if different from zero , the set $ \{\ psi , \widehat{\ psi } \} dT$ slash is linearly ds negationslash-equal independent 0 comma the curvature of alpha comma kappa comma is defined by kappa( using equiv thebar dT complex slash ds numbersbar semicolon as thus scalars comma ) dT , slash which ds = follows kappa N from comma the fact that the where N is some unit vector comma called normal of alpha comma and T and N are orthogonal to \noindent determinant of the matrix formed by the components of $ \ psi $ and $ \widehat{\ psi } $ i s

\ [ \ l e f t \arrowvert\ begin { array }{ cc }\ psi 1 & − \ psi 2 \\\ psi 2 & \ psi 1 \end{ array }\ right \arrowvert = \mid \ psi 1 \mid ˆ{ 2 } + \mid \ psi 2 \mid ˆ{ 2 }\ ]

\noindent and this sum is equal to zero if and only if $ \ psi 1 $ and $ \ psi 2 $ are simultaneously equal

\noindent to zero .

\ centerline { $ 3 . F−r { enet }$ equations }

\noindent For any curve $ \alpha :I \rightarrow R ˆ{ 3 }$ such that $ d \alpha ( t ) / dt \ne 0 , $ \quad there exists a function $ s = s ( t ) , $ \quad called the arclength , \quad such that if the curve is parametrized by $ s , $ $ \mid d \alpha / ds \mid = 1 . $ \quad Then $ , T \equiv d \alpha / ds $ is a unit vector , called the tangent of $ \alpha $ and i f $ dT / ds \not= 0 ,$ the curvature of $ \alpha , \kappa , $ is defined by $ \kappa \equiv \mid dT / ds \mid ; $ thus $ , dT / ds = \kappa N , $ where $ N $ is some unit vector , called normal of $ \alpha , $ and $ T $ and $ N $ are orthogonal to 3 0 .. G period F period TORRES DEL CASTILLO ampersand G period S A-acute sub NCHEZ BARRALES\noindent 3 0 \quad G . F . TORRES DEL CASTILLO $ \& $ G . S $ \acute{A} { NCHEZ }$ BARRALESeach other period The binormal vector comma B comma is defined by B equiv T times N period The derivative \noindentdB slash dseach is also other proportional . The to binormal N and therefore vector dB slash $ ds , = minusB , tau $ N commais defined where tau by is $Bsome \equivreal hyphenT valued\times functionN called . torsion $ The period derivative .. Using that open brace T comma N comma B closing brace$dB is an orthonormal / ds $ is also proportional to $N$ and therefore $dB / ds 3 0 G . F . TORRES DEL CASTILLO & G.S A´ BARRALES = set− comma \tau from theN foregoing , $ where relations one$ \ deducestau $ that i s dN some slashNCHEZ ds = minus kappa T plus tau B period each other . The binormal vector ,B, is defined by B ≡ T × N. The derivative .. The dB/ds is also proportional to N and therefore dB/ds = −τN, where τ is some \noindentformulas r e a l − valued function called torsion . \quad Using that $ \{ T real - valued function called torsion . Using that {T,N,B} is an orthonormal ,N,BLine 1 dT sub ds =\} kappa$ isN comma an orthonormal Line 2 dN sub ds = minus kappa T plus tau B comma Line 3 dB set , from the foregoing relations one deduces that dN/ds = −κT + τB. The subset ds = , minusfrom tau the N foregoing comma relations one deduces that $ dN / ds = − formulas \kappathat expressT the + derivatives\tau ofB T comma . $ N\quad and BThe in terms of the same vectors con hyphen formulasst itute the Frenet equations open parenthesis see comma for example comma open square bracket 6 dTds = κN, comma 7 closing square bracket closing parenthesis period dN = −κT + τB, \ [ \Accordingbegin { a lto i g the n e d results} dT presented{ ds } in= the preceding\kappads sectionN, comma\\ there exists a dN { ds } = − \kappa T + \tau B, \\ spinor comma psi comma defined up to sign commadB suchds that= −τN, dB { ds } = − \tau N, \end{ a l i g n e d }\ ] Equation: open parenthesisthat express 8 closing the derivatives parenthesis of T,N .. Nand plusB iBin = terms psi to ofthe the power same of vectors t sigma con psi - comma st T = minus psi-hatwideitute to the the Frenet power ofequations t sigma ( psi see , for example , [ 6 , 7 ] ) . with psi to the power of t psi = 1 period .. Hence comma the spinor psi represents the triad open brace \noindent that expressAccording the derivatives to the results presented of $ T in the , preceding N$ and section $ , B there $ exists in terms a of the same vectors con − N comma B comma Tspinor closing, ψ, bracedefined and up the to sign , such that stvariations itute ofthe this Frenet triad along equations the curve comma( see given , for by example the Frenet , equations [ 6 , 7comma ] ) must . correspond to some expression for d psi slash ds period \ hspace ∗{\ f i l l } According to the results presented in the preceding section , there exists a Differentiating the first equation open parenthesisN + iB 8= closingψtσψ, parenthesis T = −ψbtσψ with respect to s and using(8) the Frenet \noindentequations ones p i nfinds owith r $ψt ,ψ =\ psi 1. Hence, $ , definedthe spinor upψ represents to sign the , suchtriad { thatN,B,T } and the Equation: open parenthesisvariations of 9 closing this triad parenthesis along the .. curve minus , kappa given T by plus the tau Frenet B minus equations i tau , N must = open parenthesis\ begin { a l d i g psi n ∗} slashcorrespond ds closing to parenthesis some expression to the for powerdψ/ds. of t sigma psi plus psi to the power of t sigma openN parenthesis + iB d = psi slashDifferentiating\ psi ds closingˆ{ t }\ parenthesis the firstsigma equation period\ psi ( 8 ) with, respect T = to s−and using \widehat the Frenet{\ psi } ˆ{ t } \sigmaSince open\ psi brace\equationstag psi comma∗{$ ( one psi-hatwide finds 8 ) $ closing} brace is a basis for the two hyphen component spinors comma\end{ a there l i g n exist∗} two open parenthesis possibly complex hyphen valued closing parenthesis functions comma f and g comma such that t t \noindentd sub ds psiwith = f psi $ plus\ psi g psi-hatwideˆ{ t −}\κT +psiτB − iτN== 1 (dψ/ds .) $ σψ\+quadψ σ(dψ/dsHence). , the spinor(9) $ \ psi $ representsand substituting the this triad relation $ into\{ openN,B,T parenthesis 9 closing parenthesis\} $ commaand the using again open parenthesisvariations 8 closing parenthesis ofSince this{ commatriadψ, ψb} is we along a basis have for the the curve two - component , given spinorsby the , there Frenet exist equations two ( possibly , must correspondminus kappa toT minus somecomplex i tau expression - valuedopen parenthesis ) functions for N $, f plus dand iBg,\ psi closingsuch that parenthesis/ ds = . f $ open parenthesis N plus iB closing parenthesis minus gT plus f open parenthesis N plus iB closing parenthesis minus gT Differentiating the first equation ( 8 ) with respect to $ s $ and using the Frenet which amounts to f = minus i tau slash 2 commad gds =ψ = kappafψ + slashgψb 2 period .. The second equation in equations one finds open parenthesis 8 closingand substituting parenthesis this does relation not into ( 9 ) , using again ( 8 ) , we have give additional relations period Thus comma we have proved the following \ begin { a l i g n ∗} Proposition 1 period .... If the−κT two− iτ hyphen(N + iB component) = f(N + spinoriB) − psigT + representsf(N + iB the) − triadgT open brace N comma− \ Bkappa comma TT closing + brace\tau B − i \tau N = ( d \ psi / ds ) ôf{ at curve}\ parametrizedsigmawhich\ amountspsi by its arclength+ to f \=psi− siτ/ commaˆ{2, gt = according}\κ/2.sigmaThe to open second( parenthesis equation d \ psi in8 closing ( 8/ ) does parenthesis ds not ) comma. \ tag the∗{$ Frenet ( equations 9give ) additional $} relations . Thus , we have proved the following \endare{ a equivalent l i g n ∗} toProposition the s ingle spinor 1 . equation If the two - component spinor ψ represents the triad Equation: open parenthesis{N,B,T } 1 0 closing parenthesis .. d sub ds psi = 2 to the power of 1 open parenthesis minus\noindent i tau psiSince plus kappaof a$ curve\{\ psi-hatwide parametrizedpsi closing, by parenthesis its\widehat arclength comma{\s,psiaccording}\} to$ ( 8 )is , the a Frenetbasis equations for the two − component spinors , there exist two ( possibly complexwhere tau− andvalued kappaare equivalent ) denote functions the to torsion the s $ ingle and , curvature spinor f $ equation and of the $g curve comma , $ respectively such that period \ [ d { ds }\ psi = f \ psi + g \widehat{\ psi }\ ] 1 ddsψ = 2 (−iτψ + κψb), (10)

\noindent and substitutingwhere τ and κ thisdenote relation the torsion intoand curvature ( 9 ) of , the using curve again , respectively ( 8 ) . , we have

\ [ − \kappa T − i \tau (N+iB )=f (N+iB ) − gT + f ( N + iB ) − gT \ ]

\noindent which amounts to $ f = − i \tau / 2 , g = \kappa / 2 . $ \quad The second equation in ( 8 ) does not give additional relations . Thus , we have proved the following

\noindent Proposition 1 . \ h f i l l I f the two − component spinor $ \ psi $ represents the triad $ \{ N,B,T \} $

\noindent of a curve parametrized by its arclength $ s , $ according to ( 8 ) , the Frenet equations are equivalent to the s ingle spinor equation

\ begin { a l i g n ∗} d { ds }\ psi = 2 ˆ{ 1 } ( − i \tau \ psi + \kappa \widehat{\ psi } ), \ tag ∗{$ ( 1 0 ) $} \end{ a l i g n ∗}

\noindent where $ \tau $ and $ \kappa $ denote the torsion and curvature of the curve , respectively . S PINOR FORMULATION OF THE DIFFERENTIAL GEOMETRY OF CURVES .. 3 1 \ hspace4 period∗{\ ..f Applications i l l }S PINOR FORMULATION OF THE DIFFERENTIAL GEOMETRY OF CURVES \quad 3 1 A basic theorem of differential geometry est ablishes that if two curves in R to the power of 3 \ centerlinehave the same{4 curvature . \quad andApplications torsion functions} then comma after translating and ro hyphen tating appropriately one of them comma the two curves coincide at all their points period \noindentIn order toA prove basic this theorem theorem we of shall differential consider two spinors geometry phi and est psi ablishes such that that if two curves in $ Rphi ˆ{ to3 the}$ power of t phi = 1 and psi to the power of t psi = 1 comma then open parenthesis psi plusminuxhave the phi same closing curvature parenthesisS PINOR andto theFORMULATION torsion power of t functions OFopen THE parenthesis DIFFERENTIAL then psi , plusminux GEOMETRY after translating phi OF closing CURVES parenthesis and 3 1 ro − 4 . Applications =tating 2 plusminux appropriately psi to the power one of t ofphi themplusminux , the phi to two the curves power of coincidet psi = 2 plusminux at all psi their to the points power . A basic theorem of differential geometry est ablishes that if two curves in 3 have ofIn t phi order plusminux to prove psi to the this power theorem of t phi we = 2 shall plusminux consider 2 Re open two parenthesis spinors psi to $ the\phi powerR$ of and t phi the same curvature and torsion functions then , after translating and ro - tating closing$ \ psi parenthesis$ such period that appropriately one of them , the two curves coincide at all their points . In order to Assuming that the spinors phi and psi correspond to two curves with the same prove this theorem we shall consider two spinors φ and ψ such that \ beginfunctions{ a l i kappag n ∗} and tau comma from open parenthesis 1 0 closing parenthesis it follows that \phiLineˆ 1{ dst to}\ thephi power of= d open 1 parenthesis and \ psi psi toˆ{ thet power}\ psi of t phi= closing 1 parenthesis , then = 2\\ to( the \ psi \pm \phi ) ˆ{ t } ( \ psi \pm \phi ) = 2 \pm \ psi ˆ{ t } power of 1 open parenthesis minus i tau psi plus kappa psi-hatwide closingφtφ = 1andparenthesisψtψ = 1 to, then the power of t phi\phi plus 2\pm to the power\phi ofˆ{ 1 psit }\ to thepsi power= of t open 2 parenthesis\pm \ psi minusˆ{ i taut }\ phi plusphi kappa\pm hatwide-phi\ psi ˆ{ t } t t t t t t closing\phi parenthesis= 2 Line\pm 2 =( 2ψ to± φ the Re) (ψ power± (φ) of = 1\ 2psi open± ψ φˆ parenthesis{± φt ψ}\= 2 i±phi tauψ φ psi±). toψ φ the= power 2 ± 2Re( ofψ t phiφ). plus kappa psi-hatwide\end{ a l i g nto∗} the power of t sub phi minus i tau psi to the power of t phi plus kappa psi to the power of t Assuming that the spinors φ and ψ correspond to two curves with the same func- hatwide-phi closing parenthesis = kappa 2 open parenthesis psi-hatwide to the power of t sub phi plus psi tions κ and τ, from ( 1 0 ) it follows that to\noindent the power ofAssuming t hatwide-phi that closing the parenthesis spinors $ \phi $ and $ \ psi $ correspond to two curves with the same f u n c t i o n s $ \kappa $ and $ \tau , $ from ( 1 0 ) it follows that which is pure imaginary comma and therefore open parenthesis psi plusminuxt phi closing parenthesis to dsd(ψtφ) = 21(−iτψ + κψ)tφ + 21ψ (−iτφ + κφ) the power of t open parenthesis psi plusminux phi closing parenthesisb is a constant periodb Hence comma if \ [ \ begin { a l i g n e d } ds ˆ{ d } (1 \tpsi ˆ{t t }\t phi t ) =t 2t ˆ{ 1 } ( − the triads open brace N comma= B comma 2 (iτψ Tφ closing+ κψbφ − braceiτψ correspondingφ + κψ φb) = κ2( toψb theφ + twoψ φb curves) coincide for somei \ valuetau \ psi + \kappa \widehat{\ psi } ) ˆ{ t }\phi + 2 ˆ{ 1 which is pure imaginary , and therefore (ψ ± φ)t(ψ ± φ) is a constant . Hence , if \ psiof sˆ open{ t parenthesis} ( − whichi is\ achievedtau \ byphi appropriately} + \kappa translating\ andwidehat rotating{\phi one} of them) \\ closing the triads {N,B,T } corresponding to the two curves coincide for some value parenthesis= 2 ˆ comma{ 1 } ( i \tau \ psi ˆ{ t }\phi + \kappa \widehat{\ psi } ˆ{ t } {\phi } of s( which is achieved by appropriately translating and rotating one of them ) , − ati that\ pointtau phi\ coincidespsi ˆ{ witht }\ psi orphi with minus+ \ psikappa and this\ coincidencepsi ˆ{ t is}\ maintainedwidehat{\phi} at that point φ coincides with ψ or with −ψ and this coincidence is maintained for )for = all\ skappa period{ ..2 This} implies( \ thatwidehat the t{\ angentpsi } vectorsˆ{ t to} the{\ twophi curves} + coincide\ psi forˆ{ t }\widehat{\phi} all s. This implies that the t angent vectors to the two curves coincide for ) \allend s{ anda l i comma g n e d }\ since] for a curve alpha comma T = d alpha slash ds comma the two curves can only all s and , since for a curve α, T = dα/ds, the two curves can only differ by differ by a constant vector , but the previously mentioned translation makes the curves a constant vector comma but the previously mentioned translation makes the curves coincide for some value of s, and therefore they coincide for all s, thus proving the \noindentcoincide forwhich some value is pure of s comma imaginary and therefore , and they therefore coincide for $ all ( s comma\ psi thus\ provingpm \phi ) ˆ{ t } theorem . ( the\ psi theorem\pm period \phi ) $ is a constant . Hence , if Another application of the spinor form of the Frenet equations , ( 1 0 ) , can Another application of the spinor form of the Frenet equations comma .. open parenthesis 1 0 closing be given in the case where the quotient τ/κ is some constant . There exists parenthesis\noindent commathe tcan r i a d s $ \{ N,B,T \} $ corresponding to the two curves coincide for some value a constant angle θ(0 ≤ θ ≤ π) such that τ/κ = cot θ. Then , from ( 1 0 ) we be given in the case where the quotient tau slash kappa is some constant period .... There exists \noindenta constantof angle $have theta s that open ( $dψ/ds parenthesis which= −[iκ/ is 0(2 lessachieved sin orθ)]( equal cos by thetaθψ appropriately+ lessi sin orθψb equal) or , pi equivalently translatingclosing parenthesis, dψ/dsb and= such rotating one of them ) , thatat tau that slash point kappa[ =iκ/ $ cot(2\phi sin thetaθ$)]( period cos coincidesθψb ..+ Theni sin comma withθψ). By from $ combining\ psi open$ parenthesis these or equations with 1 0 closing $ one− finds parenthesis \ psi that$ weand this coincidence is maintained iκ f ohave r a thatl l $d psi s slash . $ dsdsd =\(quad cosminus (θ/This open2)ψ + square impliesi sin (θ/ bracket2)ψ thatb) = i− kappa the2 sin slash tθ( angent cos open (θ/2) parenthesis vectorsψ + i sin ( 2θ/to sin2) theψb theta) two clos- curves coincide for ing parenthesis closingwhich square leads bracket to open parenthesis cos theta psi plus i sin theta psi-hatwide closing i R s 0 0 parenthesis\noindent orall comma $ equivalently s $ and comma , sincecos d (θ/ hatwide-psi for2)ψ + ai curvesin slash (θ/2) dsψb $ ==\alpha exp (−2 sin,θ Tκ(s ) =ds )φ, d (11)\alpha /open ds square , $ bracket thewhere i two kappaφ is curves someslashopen spinor can parenthesis independent only differ 2 sin of thetas, bywith closingφtφ = parenthesis 1. Hence , closing square bracket a constant vector , but the previously mentioned translationi R s 0 0 makes the curves open parenthesis cos theta psi-hatwidecos (θ/ plus2)ψb i+ sini sin theta (θ/ psi2)ψ closing= exp ( parenthesis2sin θ κ( periods )ds )φb By, combining these equationscoincide one for finds some that value of $ s , $ and therefore they coincide for all $ s , $ds to thus the power proving of d open parenthesis cos open parenthesis theta slash 2 closing parenthesis psi plus i sinthe open theorem parenthesis . theta slash 2 closing parenthesis psi-hatwide closing parenthesis = minus 2 sine to the power of i kappa theta open parenthesis cos open parenthesis theta slash 2 closing parenthesis psi plus i sin open\ hspace parenthesis∗{\ f i l theta l }Another slash 2 closing application parenthesis of hatwide-psi the spinor closing form parenthesis of the Frenet equations , \quad ( 1 0 ) , can which leads to \noindentcos open parenthesisbe given theta in slashthe 2 case closing where parenthesis the quotientpsi plus i sin open $ \tau parenthesis/ theta\kappa slash$ 2 closing is some constant . \ h f i l l There exists parenthesis psi-hatwide = exp parenleftbigg minus 2 sine to the power of i theta integral to the power of s kappa\noindent open parenthesisa constant s to the angle power $ of\ primetheta closing( parenthesis 0 \ leq ds to the\theta power of prime\ leq parenrightbigg\ pi ) $ phisuch comma that open $ parenthesis\tau / 1 1\ closingkappa parenthesis= $ cot $ \theta . $ \quad Then , from ( 1 0 ) we havewhere that phi is $ some d spinor\ psi independent/ ds of s = comma− with[ phi i to the\kappa power of/ t phi ( = 1 2 period $ s i Hence n $ \theta comma) ] ( $ cos $ \theta \ psi + i $ s i n $ \theta \widehat{\ psi } ) $cos oropen , parenthesis equivalently theta slash $ 2 , closing d parenthesis\widehat psi-hatwide{\ psi } plus/ i sin ds open = parenthesis $ theta slash 2 closing$ [ parenthesis i \kappa psi = exp/ parenleftbigg ( 2 $ sub s i n 2 sine $ \ totheta the power) of i ] theta ( integral $ cos to the $ \ powertheta of s kappa\widehat open{\ parenthesispsi } + s to the i $ power s i n of prime $ \theta closing parenthesis\ psi ds) to the . $ power By of combining prime parenrightbigg these equations one finds that hatwide-phi sub comma \ centerline { $ ds ˆ{ d } ( $ cos $ ( \theta / 2 ) \ psi + i $ s i n $ ( \theta / 2 ) \widehat{\ psi } ) = − 2 \ sin ˆ{ i \kappa } \theta ( $ cos $ ( \theta / 2 ) \ psi + i $ s i n $ ( \theta / 2 ) \widehat{\ psi } ) $ }

\noindent which leads to

\ hspace ∗{\ f i l l } cos $ ( \theta / 2 ) \ psi + i $ s i n $ ( \theta / 2 ) \widehat{\ psi } = $ exp $ ( − 2 \ sin ˆ{ i }\theta \ int ˆ{ s } \kappa ( s ˆ{\prime } ) ds ˆ{\prime } ) \phi , ( 1 1 ) $

\noindent where $ \phi $ is some spinor independent of $ s , $ with $ \phi ˆ{ t } \phi = 1 . $ Hence ,

\ centerline { cos $ ( \theta / 2 ) \widehat{\ psi } + i $ s i n $ ( \theta / 2 ) \ psi = $ exp $ ( { 2 }\ sin ˆ{ i }\theta \ int ˆ{ s } \kappa ( s ˆ{\prime } ) ds ˆ{\prime } ) \widehat{\phi} { , }$ } 3 2 .. G period F period TORRES DEL CASTILLO ampersand G period S A-acute sub NCHEZ BARRALES\noindent 3 2 \quad G . F . TORRES DEL CASTILLO $ \& $ G . S $ \acute{A} { NCHEZ }$ BARRALESwhich implies that whichpsi = exponent implies parenleftbigg that minus 2 sine to the power of i theta integral to the power of s kappa open parenthesis s to the power of prime closing parenthesis ds to the power of prime parenrightbigg cosine open parenthesis\ begin { a l theta i g n ∗} slash 2 closing parenthesis phi Equation: open parenthesis 1 2 closing parenthesis .. minus i exponent\ psi = parenleftbigg\exp sub( 2 sine− to2 the power\ sin ofˆ{ i thetai }\ integraltheta to the\ powerint ofˆ{ s kappas }\ openkappa parenthesis( 3 2 G . F . TORRES DEL CASTILLO & G.S A´ BARRALES which implies that ss to ˆ{\ theprime power of} prime) closing ds ˆ{\ parenthesisprime } ds to) the\ powercos of( primeNCHEZ\ parenrightbiggtheta / sine 2 open ) parenthesis\phi \\ − thetai \ slashexp 2 closing( { parenthesis2 }\ sin phi-hatwideˆ{ i }\theta \ int ˆ{ s }\kappa ( s ˆ{\prime } ) ds ˆ{\prime } ) \ sin ( \theta Z/s 2 ) \widehat{\phi}\ tag ∗{$ ( and comma therefore comma i 0 0 1psi-hatwide 2 ) $} = minus i exponent parenleftbiggψ = exp(− minus2 sin θ 2 sineκ(s to)ds the) cos( powerθ/2)φ of i theta integral to the \end{ a l i g n ∗} power of s kappa open parenthesis s to the power of primeZ closings parenthesis ds to the power of prime i 0 0 parenrightbigg sine open parenthesis theta slash−i exp( 2 closing2sin θ parenthesisκ(s )ds phi) sin( Equation:θ/2)φb open parenthesis(12) 1 3 closing\noindent parenthesisand .. , plus therefore exponent parenleftbigg, sub 2 sine to the power of i theta integral to the power of s kappa open parenthesisand ,s thereforeto the power , of prime closing parenthesis ds to the power of prime parenrightbigg cosine\ begin open{ a l parenthesisi g n ∗} theta slash 2 closing parenthesis phi-hatwide sub comma \widehatthen comma{\ psi according} = to open− parenthesisi \exp 8 closing( parenthesis− 2 comma\ sin ˆ denoting{ i }\ bytheta open brace\ aint commaˆ{ s } \kappa ( s ˆ{\prime } ) ds ˆ{\prime Z} s ) \ sin ( \theta / 2 b comma c closing brace the triad of constantψb = − uniti exp( vectors−2 sini θ κ(s0)ds0) sin(θ/2)φ ) corresponding\phi \\ + to the\exp spinor phi( open{ 2 parenthesis}\ sin ˆ i{ periodi }\ e periodtheta comma\ int a plusˆ{ i bs =}\ phi tokappa the power( s ˆ{\prime } ) ds ˆ{\prime } ) \cos Z s( \theta / 2 ) \widehat{\phi} { , }\ tag ∗{$ ( of t sigma phi comma c = minus phi-hatwide to the poweri of t sigma0 phi0 closing parenthesis comma + exp(2sin θ κ(s )ds ) cos(θ/2)φb, (13) 1T 3= cos ) theta $} c plus sin theta bracketleftbigg sin parenleftbigg sub sine to the power of 1 theta integral to\end the{ powera l i g n ∗} of s kappa open parenthesis s to the power of prime closing parenthesis ds to the power of prime parenrightbiggthen a , according to ( 8 ) , denoting by { a , b , c } the triad of constant unit \noindentEquation:then open parenthesisvectors , according 14 closing to ( parenthesis 8 ) , denoting .. minus cosine by \{ parenleftbigga , b , sub c \} sinethe to the triad power of of constant unit vectors t t 1 theta integral to thecorresponding power of s kappa to the open spinor parenthesisφ( i . e . s , toa the+i b power= φ ofσφ, primec = closing−φb σφ parenthesis), ds to \noindent corresponding to the spinor $ \phi 1($R s i.e.,a$+0 0 i$b the power of prime parenrightbigg b bracketrightbiggT = cos θ c + period sin θ[ sin (sinθ κ(s )ds ) a $ =This\ lastphi equationˆ{ t }\ provessigma the validity\phi of the following, $ c $ = − \widehat{\phi} ˆ{ t }\sigma \phiProposition) ,.... $ 2 period .... The .... tangent vector of aZ ....s curve .... with tau slash kappa .... constant 1 0 0 forms .... a − cos(sinθ κ(s )ds )b]. (14) \ centerlineconstant angle{ $ open T parenthesis = $ cos theta $ \ closingtheta parenthesis$ c $ + with $ a s constant i n $ \ vectortheta open[ parenthesis $ s i n c$ ( {\ sin }ˆ{ 1 } closing\theta parenthesis\ int ˆ period{Thiss last}\ equationkappa proves( the s validity ˆ{\prime of the} following) ds ˆ{\prime } ) $ a } Recalling that TProposition = d alpha slash 2 .dsThe comma tangent integrating vector open of a parenthesis curve with 1τ/κ 4 closingconstant parenthesis forms a with respect\ begin to{ a s l onei g n obtains∗} constant the angle (θ) with a constant vector ( c ). −expression \cos for( any{\ curveRecallingsin of} thisˆ{ that1 class}\T = opendα/ds,theta parenthesisintegrating\ int calledˆ ({ 1 cylindricals 4 )}\ withkappa respect helices to open(s one square s obtains ˆ{\ bracketprime the 6 } comma) ds 7 ˆclosing{\prime squareexpression} bracket) for closing b any ] curve parenthesis . of\ tag this period∗{ class$ ( ( called 14 cylindrical ) $} helices [ 6 , 7 ] ) . \endThe{ a preceding l i g n ∗} equationsThe are preceding also valid equationswhen tau =are 0 also comma valid making when τ theta= 0, =making pi slashθ = 2π/ semicolon2; equa -equa hyphen tion ( 1 4 ) reduces then to \noindenttion open parenthesisThis lastT 1= 4 equation sin closing (R s κ parenthesis(s0 proves)ds0) a − reduces thecos ( validityR s thenκ(s0) tods0) b of, the ( 1following 5 ) which shows explicitly T = sin parenleftbiggthat theintegral curve to lies the on power a plane of s ( kappa normal open to c parenthesis) and allows s to the power of prime closing parenthesis\noindent dsProposition to theus power to find of\ the primeh f parametrici l l parenrightbigg2 . \ formh f i l of l a anyThe minus plane\ h cos f curve i l parenleftbigg l tangent given its curvature integral vector to ( of cf the . a [ power 7\ ]h , f p i of l l s curve \ h f i l l with kappa$ \tau open parenthesis/ \kappa. s to$ the\ powerh f i l lof primeconstant closing forms parenthesis\ h f i ds l l toa the power of prime parenrightbigg b comma .. open parenthesis38 ) . For 1 instance 5 closing , parenthesis in the simple case where κ is constant , equation ( 1 5 ) gives \noindentwhich showsconstant explicitly angle that the $curve ( lies\theta onT = a planesin (κs) open) $a − withparenthesiscos ( aκs) constantb normal to vector c closing ( parenthesis c ) . and allows and integrating this expression with respect to s one finds that \ hspace ∗{\ f i l l } Recalling that $T = d \alpha / ds , $ integrating ( 1 4 ) with respect to us to find the parametricα(s) = 1κ form(− cos of ( anyκs) planea − sin curve (κs) givenb )+ itsd curvature, where d openis a parenthesis constant vector cf period , thus open square$ s $ bracket one 7obtains closingshowing square the that bracket the curve comma is a p circle period of radius 38 closing parenthesis period For instance comma in the simple case where kappa is constant comma equation\noindent openexpression parenthesis 1 5 for closing any parenthesis curve of gives this class ( called cylindrical helices [ 6 , 7 ] ) . T = sin open parenthesis kappa s closing parenthesis1 a/κ. minus cos open parenthesis kappa s closing parenthesis\ hspace ∗{\ b f i l l }The preceding equations are also valid when $ \tau = 0 , $ makingand integrating $ \theta this expression= \ pi with/ respect 2 to s ; one $ finds equa that− alpha open parenthesis s closing parenthesis = 1 kappa open parenthesis minus cos open parenthesis kappa\noindent s closingtion parenthesis ( 1 4 a ) minus reduces sin open then parenthesis to kappa s closing parenthesis b closing parenthesis plus d comma $where T = d is $ a constant s i n $ vector ( \ commaint ˆ{ thuss showing}\kappa that the( curve s is ˆ a{\ circleprime of radius} ) ds ˆ{\prime } ) $1 slash a kappa$ − $ period cos $ ( \ int ˆ{ s }\kappa ( s ˆ{\prime } ) ds ˆ{\prime } ) $ b , \quad ( 1 5 ) which shows explicitly that the curve lies on a plane ( normal to c ) and allows

\noindent us to find the parametric form of any plane curve given its curvature ( cf . [ 7 ] , p .

\noindent 38 ) . For instance , in the simple case where $ \kappa $ is constant , equation ( 1 5 ) gives

\ centerline { $ T = $ s i n $ ( \kappa s ) $ a $ − $ cos $ ( \kappa s ) $ b }

\noindent and integrating this expression with respect to $ s $ one finds that

\noindent $ \alpha ( s ) = 1 {\kappa } ( − $ cos $ ( \kappa s ) $ a $ − $ s i n $ ( \kappa s ) $ b $ ) + $ d , where d is a constant vector , thus showing that the curve is a circle of radius

\ begin { a l i g n ∗} 1 / \kappa . \end{ a l i g n ∗} S PINOR FORMULATION OF THE DIFFERENTIAL GEOMETRY OF CURVES .. 33 \ hspaceAs pointed∗{\ f out i l l in}S Sec PINOR period FORMULATION 2 comma there isOF a twoTHE hyphen DIFFERENTIAL to hyphen one GEOMETRY homomorphism OF CURVES of the group\quad 33 SU open parenthesis 2 closing parenthesis .. onto the rotation group in three dimensions SO open parenthesisAs pointed 3 closing out in parenthesis Sec . 2period , there .. Assuming is a that two − to − one homomorphism of the group SUpsi ( to 2 the ) \ powerquad ofonto t psi = the 1 comma rotation as in open group parenthesis in three 8 closing dimensions parenthesis SO and ( 3 open ) . parenthesis\quad Assuming that 1 0 closing parenthesis comma the 2 times 2 matrix \noindentQ equiv Row$ 1\ psipsi 1 psi-hatwideˆ{ t }\ 1psi Row 2= psi 2 1 hatwide-psi ,$ asin(8)and(10),the2 . $2 \timesbelongs to2 SU $ open matrix parenthesisS PINOR 2 FORMULATION closing parenthesis OF THE and DIFFERENTIAL corresponds GEOMETRY to the rotation OF CURVES that carries 33 the As pointed out in Sec . 2 , there is a two - to - one homomorphism of the group canonical SU ( 2 ) onto the rotation group in three dimensions SO ( 3 ) . Assuming that \ [Qbasis into\equiv the orthonormal\ l e f t (\ begin basis open{ array brace}{ cc N}\ commapsi B comma1 & T\ closingwidehat brace{\ psi period} ..1 Making\\\ usepsi ψtψ = 1, as in ( 8 ) and ( 1 0 ) , the 2 × 2 matrix of2 open & \ parenthesiswidehat{\ 6psi closing} parenthesis2 \end{ array comma}\ openright parenthesis) \ ] 7 closing parenthesis comma and open parenthesis 1 0 closing parenthesis comma ! ψ1 ψ1 we find that the Frenet equations take the form Q ≡ b \noindentEquation:belongs open parenthesis to SU 1 6( closing 2 ) and parenthesis corresponds .. dQ dsψ2 = to QHψb2 the open rotation parenthesis that s closing carries parenthesis the canonical comma with H openbelongs parenthesis to SU s ( closing 2 ) and parenthesis corresponds equiv to the minus rotation 2 to that the carries power theof i canonical Row 1 tau open parenthesis\noindent s closingbasis parenthesis into the minus orthonormal i kappa open basis parenthesis $ \{ s closingN,B,T parenthesis Row 2 i kappa\} open. $ \quad Making usebasis of into ( 6 the ) orthonormal , ( 7 ) , basis and{N,B,T ( 10} ). ,Making use of ( 6 ) , ( 7 ) , and ( 1 parenthesis s closing0 parenthesis ) , we find minus that the tau Frenet open parenthesis equations take s closing the form parenthesis . period weThere find are that two cases the inFrenet which theequations integration take of open the parenthesis form 1 6 closing parenthesis is relatively simple semicolon the \ beginfirst one{ a l correspondsi g n ∗} to the trivial case where H does not depend on sτ open(s) parenthesis−iκ(s) and dQ = QH(s), with H(s) ≡ −2i . (16) dQthe{ solutionds } of=QH open parenthesis ( 1 sds 6 closing ) parenthesis , with is given H by ( Q openiκ s( parenthesiss) ) −τ(\sequiv) s closing parenthesis− 2 ˆ{ i }\ l e f t (\ begin { array }{ cc }\tau =( Q open s parenthesis ) & − 0 closingi \kappa parenthesis( exp open s parenthesis ) \\ i sH\kappa closing parenthesis( s closing ) & parenthesis− \tau ..( and s the second ) \end one{Therearray are}\ right two cases). in\ whichtag ∗{ the$ ( integration 1 6 of ( ) 1 6$} ) is relatively simple ; the \endcorresponds{ a l i g n ∗} to thefirst case one where corresponds H open parenthesis to the trivial s closing case parenthesis where H does commutes not depend with H on opens( and parenthesis the s to the power of primesolution closing of parenthesis ( 1 6 ) is given for all by sQ to(s the) power = Q of(0) prime exp (periodsH)) .. and This the last second one 0 0 \noindentcondition amountsTherecorresponds are to tau two open cases to parenthesis the case in whichwhere s closingH the(s parenthesis) commutes integration slash with kappaH of(s ) ( open for 1 all 6 parenthesis )s . isThis relatively s last closing simple ; the 0 0 parenthesisfirst one = tau corresponds opencondition parenthesis amounts to s the to the to trivialτ power(s)/κ( ofs) caseprime = τ(s closing) where/κ(s ) parenthesis and $ Hthe $ solution slash does kappaof not ( 1 6 open depend ) can parenthesis be on $ s s( to $ the and power of primeexpressed closing as parenthesis and the solution of open parenthesis 1 6 closing parenthesis can thesolutionof(16)isgivenby $Q ( s ) = Q ( 0 )$ exp be s $ ( sH ) ) $ \quad and the second one Z expressed as Q(s) = Q(0) exp H(s0)ds0, correspondsQ open parenthesis to the s closing case parenthesis where $H = Q open ( parenthesis s )0 $ 0 closing commutes parenthesis with exponent $H (integral s ˆ{\prime } sub)$ 0 to for the all power $sˆ ofwhich s H{\ open is equivalentprime parenthesis} to. the s $ to solution the\quad power givenThis of by prime l( a 1 s 2t closing ) and parenthesis( 1 3 ) . ds to the power of primecondition comma amounts to $ \tau (5 s . Final ) / remarks\kappa ( s ) = \tau (which s ˆ{\ is equivalentprimeThe} to thefact)/ solution that the\ givenkappa Frenet by equations open( parenthesis s reduce ˆ{\prime 1to 2 a closing single} parenthesis spinor) $ equation and and the open [ equation solution parenthesis of ( 1 6 ) can be 1 3 closing parenthesis( 1 period 0 ) ] , equivalent to the three usual vector equations , is a consequence of \noindent5 period ..expressed Final remarksthe relationship as between spinors and orthogonal triads of vectors and to the use The fact that theof Frenet complex equations quantities reduce . It to may a single be noticed spinor equation that the open derivations square bracketpresented equation in the \ [open Q parenthesis ( spreceding ) 1 0closing = section Q parenthesis ( are not 0 closing a“ ) word square\exp by word bracket\ ”int translation commaˆ{ s } equivalent of{ the0 corresponding} toH the ( three ones s usual ˆ{\prime } vector) ds equations ˆ{\prime commain} the is a vector, consequence\ ] formalism of the but there exist an independent procedure in the spinor relationship betweenformalism spinors which and orthogonal is very useful triads as shown of vectors by the and result to the given use in ( 1 4 ) . of complex quantitiesIt period may also It may be noticedbe noticed that that it has the notderivations been necessary presented to in have the an explicit ex- \noindentpreceding sectionwhichpression are is not equivalent a for quotedblleft the spinor to wordcorresponding the by solution word quotedblrightto the given triad { byN,B,T translation ( 1} 2of )aof given andthe corresponding curve ( 1 3 . ) . ones in the vectorAcknowledgements formalism but there exist . The an independent second author procedure acknowledges in the the support from the \ centerlinespinor formalism{5 . Consejo which\quad is EstatalFinal very useful de remarks Ciencia as shown y} Tecnolog by the´ı resulta del Estadogiven in de open Puebla parenthesis( Me ´ xico 1 4 ) .closing parenthesis period References \noindent The fact that the Frenet equations reduce to a single spinor equation [ equation It may also be noticed that[ 1 ] it hasE . not Cartan been necessary, The to theory have anof spinors explicit , Hermann , Paris , 1 966 . ( expression for theDover spinor , New corresponding York , to the triad open brace N comma B comma T closing brace of \noindent ( 1 0 ) ] , equivalent to the three usual vector equations , is a consequence of the a given curve period reprinted 1 98 1 . ) relationship between spinors and orthogonal triads of vectors and to the use Acknowledgements[ 2 ] periodW . T The . Payne second, authorElementary acknowledges spinor theory the, support Am . Jfrom . of Phys the . 20 ( 1 952 ) 253 . of complex quantities . It may be noticed that the derivations presented in the Consejo Estatal de Ciencia[ 3 ] yG Tecnolog . F . Torres acute-dotlessi del Castillo a del Estado, 3 - D deSpinors Puebla , Spin open - Weighted parenthesis Functions M e-acute preceding section are not a ‘‘ word by word ’’ translation of the corresponding xico closing parenthesisand period their Applications , Birkh a¨ user , Boston , 2003 . onesReferences in the vector formalism but there exist an independent procedure in the spinoropen square formalism bracket 1which closing is square very bracket useful .. E as period shown .. Cartanby the comma result .. The given theory in of ( spinors 1 4 ) . comma Hermann comma Paris comma 1 966 period open parenthesis Dover comma New York comma Itreprinted may also 1 98 be 1 period noticed closing that parenthesis it has not been necessary to have an explicit expressionopen square bracket for the 2 closing spinor square corresponding bracket .. W period to the T period triad Payne $ \{ commaN,B, Elementary spinor Ttheory\} comma$ of Am a period given J period curve of . Phys period 20 open parenthesis 1 952 closing parenthesis 253 period open square bracket 3 closing square bracket .. G period F period Torres del Castillo comma 3 hyphen D\noindent Spinors commaAcknowledgements Spin hyphen Weighted . The Functions second and author their acknowledges the support from the ConsejoApplications Estatal comma de Birkh Ciencia dieresis-a y user Tecnolog comma Boston $ \acute comma{\imath 2003 period} $ a del Estado de Puebla ( M $ \acute{e} $ x i c o ) .

\ centerline { References }

\ hspace ∗{\ f i l l } [ 1 ] \quad E. \quad Cartan , \quad The theory of spinors , Hermann , Paris , 1 966 . ( Dover , New York ,

\ centerline { reprinted 1 98 1 . ) }

\ centerline { [ 2 ] \quad W. T . Payne , Elementary spinor theory , Am . J . of Phys . 20 ( 1 952 ) 253 . }

[ 3 ] \quad G . F . Torres del Castillo , 3 − D Spinors , Spin − Weighted Functions and their Applications , Birkh $ \ddot{a} $ user , Boston , 2003 . 34 .. G period F period TORRES DEL CASTILLO ampersand G period S A-acute sub NCHEZ BARRALES\noindent 34 \quad G . F . TORRES DEL CASTILLO $ \& $ G . S $ \acute{A} { NCHEZ }$ BARRALESopen square bracket 4 closing square bracket .. D period H period Sattinger ampersand O period L period Weaver comma Lie Groups and Algebras with Applications \ hspaceto Physics∗{\ f comma i l l } [ Geometry 4 ] \quad andD Mechanics . H . Sattinger comma Springer $ hyphen\& $ VerlagO . L comma . Weaver New York , Lie comma Groups and Algebras with Applications 1 986 period \ centerlineopen square{ bracketto Physics 5 closing , square Geometry bracket and .. H Mechanics period Goldstein , Springer comma Classical− Verlag mechanics , New comma York , 1 986 . } 34 G . F . TORRES DEL CASTILLO & G.S A´ BARRALES 2 nd ed period comma Addison hyphen Wesley comma Reading commaNCHEZ Mass period comma [ 4 ] & , Lie Groups and Algebras with \ hspace1 980 period∗{\ f i l l } [ 5 ] \quadD .H H . Sattinger Goldstein ,O Classical. L . Weaver mechanics , 2 nd ed . , Addison − Wesley , Reading , Mass . , Applications open square bracket 6 closing square bracket .. B period .. O quoteright Neill comma Elementary to Physics , Geometry and Mechanics , Springer - Verlag , New York , 1 986 . differential\ centerline geometry{1 980 comma . } 2 nd ed period comma Academic Press comma San [ 5 ] , Classical mechanics , 2 nd ed . , Addison - Wesley , Diego comma 1 997 period H . Goldstein Reading , Mass . , \ hspaceopen square∗{\ f i bracket l l } [ 6 7 ] closing\quad squareB. bracket\quad ..O J ’ period Neill Oprea , Elementary comma Differential differential geometry and geometry its , 2 nd ed . , Academic Press , San 1 980 . applications comma Prentice hyphen Hall comma Upper Sad hyphen [ 6 ] , Elementary differential geometry , 2 nd ed . , \ centerlinedle River comma{ Diego N period , 1 997 J period . } commaB . 1O 997 ’ Neill period Academic Press , San open parenthesis Reci b i do e n febrero d e 2004 closing parenthesis Diego , 1 997 . \ hspaceDepartamento∗{\ f i l l de} [ F 7 acute-dotlessi ] \quad J sica . MatemOprea acute-a , Differential tica geometry and its applications , Prentice − Hall , Upper Sad − [ 7 ] , Differential geometry and its applications , Prentice - Hall , Instituto de Ciencias J . Oprea Upper Sad - \ centerlineUniversidad{ Autdle o-acute River noma ,N. de Puebla J . , 1 997 . } dle River , N . J . , 1 997 . Apartado postal 1 1 52 ( Reci b i do e n febrero d e 2004 ) \ centerline7200 1 Puebla{( comma Reci bM e-acutei do e xico n febreroperiod d e 2004 ) } ´ı a´ e hyphen mail : gtorresDepartamento at f c fm period de F buapsica period Matem mx tica Instituto de Ciencias Univer- o´ e´ DepartamentoFacultad de Ciencias desidad F F $ Aut acute-dotlessi\acutenoma{\imath de s ico Puebla Matem} $ sApartado acute-a i c a Matem ticas postal $ 1\acute 1 52 7200{a} 1$ Puebla t i c a , M InstitutoUniversidad de Aut Ciencias o-acutexico . noma de Puebla e - mail : @ UniversidadApartado postal Aut 1 1 52 $ \acute{o} $ noma de Puebla gtorres f c fm . buap . mx ´ı a´ o´ Apartado7200 1 Puebla postal comma 1Facultad 1M 52e-acute de xico Ciencias period F s ico Matem ticas Universidad Aut noma e´ 7200e hyphen 1 Puebla mail : gsde , toM Puebla the $ power\acute Apartado of{ a-ne} chez$ postal at x i f c cfm1o 1 . 52 period 7200 buap 1 Puebla period , mx M xico . e - mail : gsa−n chez @ f cfm . buap . mx \ hspace ∗{\ f i l l }e − mail : gtorres $@$ f c fm . buap . mx

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