twistors and nearly autoparallel maps

8/14/2019 Twistors and Nearly Autoparallel Maps

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Vol. 7 (1996) REPORTS ON ~ATH~~ATICAI, PHYsIfS No.

WSTORS AND NEARLY AUTOP~LEL MAPSSERGIU I. VACARU

Institute of Applied Physics, Moldovan Academy of Sciences, 5 Academiei str., Chiginau, 2028Republic of Moldova (e-mail: [email protected])and

SERGIU V. OSTAFDepartment of Mathematics and Physics, Tiraspol State University, 5 Iablochkin str., Chi$&u, 2062Republic of Moldova

(Received December 6, 1994)

The purpose of the present paper is to investigate the problem of definition of twist-ors on generic curved spaces. Firstly, we consider nearly geodesic (autoparallel) maps of(pseudo)-Rieman~an spaces. Secondly, we shall define nearly autoparallel hvistor equationswhich are compatible on nearly ~nfo~aily Aat spaces. Finally, we shall study nearly auto-parailel twistor structures generating curved spaces and vacuum Einstein spaces.

1. Introduction: Spinors and maps of curved spaces with deformation of connectionsOur geometrical constructions will be realized on pairs of 4-dimensional (pseudo)-

Riemannian spaces (V, IL) with the signature (- + + +) and i-l local maps ofspaces f: V + g given by functions f(x) of class CT(U) (r > 2, T = w for analyticfunctions) and the inverse functions f?) with co~esponding non-zero Jacobians atevery point x = {x} c U c V and : = {@} c U c IL (U and g are open regions).We shall attribute the regions U and u to a common, for a given f-map, coordinatesystem, when every point q E U with coordinates d(q) is mapped onto a point q c Uwith the same coordinates xp = c&(q) = x(q), i.e. f: d(q) -+ x(q). We note thatall calculations in this work will be ha1 and will refer to common coordinates, forgiven f-maps, on open regions of spaces into considerations.The metric tensor, the connection, differential operator and the tetrads (frames)are denoted on U as g~~(x),~~~(x), I), and h;(x), respectively, where

s/N(x) = qxY$M?as, v,b = const, (1)and on u as gPy(x)=gPy(x), &~(x)+~(x),QP = D, and &t(x) = h;(x), where--

(2)(the Greek indices ~1,v, p,y, . . . range from 0 to 3). Throughout the present paper-


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310 S. I. VACARU and S. V. OSTAFwe shall use the terminology and definitions of the abstract and coordinate tensorand spinor index formalisms described in the monographs [l, 21. For simplicity, weshall write the Latin symbols a, b,a,b, . . . for both the abstract and the tetrad indicesimplying that in the first case Latin indices are abstract labels and in the secondcase (for decompositions on tetrads) we shall specify their explicit values (a, b, . . .= 0, 1,2,3). We consider spinor decompositions of metrics (1) and (2):

S&AU cpAA(Z)~,BB(Z)E~~t~,~,, (3)where O,(X) = h;(~)~,A~,cr~~ = cons& are the Infeld-van der Waerden coeffi-cients and EAB = -CBA, EA~B/= -EB/A~ A, B , A, B = 0,l) are spinor metrics

9 =fl-P _;A(x)G?fB(x)~AB~AtB5 (4 )where azA(z) = O,(X) = $(z)a,AA; if necessary, we shall write, for example,A,A A-EAB YEA ~B/~& ,W or 5:: in order to point out that these spin-tensor values areassociated to the spinor decomposition (4) on the space E.

For mutual transformations of the tensor and spinor indices one introduces theinverse Infeld-van der Waerden coefficients $J,,(z) and azA,(z), for example, B= GA , BAA and &B, = AE~EB,.Covariant derivation of spinors on V is defined in terms of the spin coefficientsY &B and r~$?~, :

DAA~ -- azA ,(~)D,eB = ~AA J


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TWISTORS AND NEARLY AUTOPARALLEL MAPS 311and, as a consequence of formulae (5), deformations of the spin coefficients, forexample,

r,,(x) = Y,S) + *Y;&)J (7)where P$(s) and *r,, = P..? - ~(&B,~pc$ - nk ,,~?,oz~) are called thePBCdeformation tensor and the deformation spin tensor, respectively. Deformations ofthe covariant derivation operator, caused by splittings of type (6) or (7) will bedenoted as

& = D, + *Dll, DA.& = DAA + *DAA. (8)In a particular case of conformal maps c: U + U, when

9--ab = Q29atl, CAB = acAB, tAB - f%AB,L?(cc) is a nonzero real function on U,

*D, = fT D,fi = D&R. (9)Conformal transforms are largely used, for example, in the twistor [2] and conformalfield theories.

We note that there are classes of l-l local maps with the deformation of con-nection ((6) or (7)) more general than that for the conformal maps (9) (see, forexample, (r~ - 2)-projective spaces [3], nearly geodesic maps (ng-maps [4]) and nearlyautoparallel maps (na-maps) of spaces with torsion and nonmetricity [5, 61, of fibrebundles [7] and of Finsler and Lagrange spaces [S]). In our papers [5, 9-111 we haveproposed to apply the ng- and na-maps for the definition of conservation laws on thecurved spaces. Na-maps were used for definition of the nearly autoparallel twistorsin connection to a possible twistor-gauge interpretation of vacuum gravitational fields[7, 1 -141.

The second objective is the investigation of na-map deformations of the twistorequations [2] (for our purposes written on space v)DcAu B) = f(&,gB + &g) = 0,-A-- (10)

where ( ) denotes symmetrization.Because for the uncharged twistors we have

whereD A(C&Jr) = _,+r,C$Q~, (11)

-abcd --ABCD~AJB fkDt=@ + EA~Lv~~LI~&~BSCLI (12)is the conformal Weyl tensor on space v, there is a hard compatibility conditionfor twistor equations (IO), namely, g_ABcDg D = 0, which characterizes, for example,conformally flat spaces. That is why a mathematically rigorous, and generally accepteddefinition of twistors was possible only for conformally flat spaces and this fact isthe main impediment to the twistor interpretation of general gravitational fields (fordetails see [2]).


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312 S. I. VACARUand S. . OSTAFOur main idea [ll, 7, 13, 141 was to define twistors not on the generic curved

spaces V, where twistor equations are incompatible, but to remove the problem onauxiliary conformally flat (or more simply, flat M) background spaces V, interrelatedwith the fundamental space-time V by means of chains of na-maps (nearly confor-ma1 maps, nc-maps). On the space M, twistor equations (10) become compatible; wecan define twisters in a standard manner as pairs of spinors, z* = (gA,7_rA). Then,transferring z* on V, by using nc-maps, we can define nearly autoparallel twistors,na-twistors, as satisfying the na-twistor equations, being the na-images of equations(10). For simplicity, in this paper we shall restrict ourselves only to the nearly geo-desically flat (ng-flat) spaces V which admit ng-maps to the Minkowski space M.We shall analyse conditions when na-twistor equations contain information on thevacuum Einstein fields.

2. Nearly geodesic maps and spinorsThe aim of this section is to present a brief introduction to the geometry of

ng-flat spaces. We shall specify basic ng-map equations and invariant conditions [4]to the case of the vacuum gravitational fields on V. Proofs are mechanical, but, inmost cases, calculations are rather tedious, and similar to those presented in [4, 131.They are omitted.2.1. Definition of ng-maps

Let us parametrize curves on U c V by functions ZY = z?(n), nr < 71< 112,withthe corresponding tangent vector field defined as up = F.

DEFINITION 1. A curve 1 is called a geodesic on V if its tangent vector fieldsatisfies the autoparallel (a-parallel) equations

UD@ = u%pu = p(q)@, (13)where ~(77) is a scalar function.

We note that for (pseudo)-Riemannian spaces the extremal curves, the geodesics,coincide with the straightest curves, a-parallels, and that is why we shall use the termgeodesics for both classes of curves (for spaces with locally isotropic or anisotropictorsion and nonmetricity we have started with a-parallel equations [5-81).On the space v we consider a new class of curves: Let a curve 1 c v be givenparametrically as zQ = zY(rj),nr < 9 < QZ,U* = s#O. We say that a 2-dimensionaldistribution E$) is coplanar along 4 if in every point 1c E 1 there is defined a%-dimensional vector space I&(z) c T,y (T,V is the tangent space at x E V) andevery vector p%$$ c E2(I), ~(0, E I, is contained in the same distribution afterparallel transports along I, i.e. p*(zp(v)) c: IQ(l).

DEFINITION 2. A curve [ is called a nearly geodesic on space E if along i there isdefined a coplanar distribution I??&) containing the vector field ~~(7) tangent to 1.Ng-maps are introduced [4] according to the


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TWISTORS AND NEARLY AUTOPARALLEL MAPS 313DEFINITION 3. Nearly geodesic maps (ng-maps) are focal l-l mappings of (pseudo)-

Riemannian spaces, ng: V-v, changing every geodesic 1 on V into a nearly geodesic1 on y.

Let a geodesic 1 c U be given by functions xa = F(n), ua = T, ~1 < 7 < 772,satisfying equations (12). We suppose that to the geodesic 1 there corresponds anearly geodesic i c u given for a chosen ng-map by the same parametrization in acommon local coordinate system on U and u. This requirement is satisfied if andonly if vectors zF, g;, = uQF and U+ = U&J;) are linearly depended in everypoint 2 E U, i.e.,

UC, = a(77F + b(n)u&for some scalar functions ~~(17) nd b(n). Putting splitting (6) into expressions for 2;)and I& on U, and from the just presented linear dependence we obtain

u~uy u6(DpP~ ~ + P&?P$) = bUy U6P$ + aua > (14)where b(q, U) = 6 - 3p and

a(% u> = C?+ bp - ubabp - p2 (15)are called the deformation parameters of the ng-maps.2.2. Classification of ng-maps

Ng-maps were classified [4] by considering the possible polynomial dependencieson 2~~ of deformation parameters (15). We shall consider the maps ng : V --f r satis-fying the reciprocity conditions (ng-: V --f V is also an g-map). This requirementis fulfilled if

P&P;; = dCqP+; + c&$),for a vector de, and a tensor cap on V.

THEOREM 1.Four classes of the ng-maps are characterized by the correspondingparamett iat ions of the deformat ion tensors and basic equat ions:

-for the t rivial ng-maps, t he geodesic maps (or 7r(o)-maps)P$W = ~(P~,*,~ (16)

w here 6p is Kronecker symbol and +p = $0(x) is a covariant vector field;-for the 7r(l)-maps, Pi;(x) is the solut ion of the equat ions

3D,P$ = 2Riz ,r,a - 214i,ra, + 6bcc,P&j + 6a(,&); (17)-for the n(2)-maps,

Pi; = 24(&, + 2cr(,F& (18)w here F; = F;(x) is such t hat Ft = F; = e@ (e = fl):

8,pF,iFx - d,xF,jF; = 0 (19)


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314 S. I. VACARU and S. V. OSlXF([ ] denotes antisymmetrization) and solves the equations

D,jF, + pL yF$Y ; - pQF; = 0 (20)for a covarian t vector field pL r = &x);

-for th e n(3)-m aps,P$- = 2&S ,; + Oy6(PT , (21)

wh ere th e cont rava rian t vector field yT = q(x) sati sjies th e equat ionsD,cpp = uS ,p + pL ,cpB, (22)

for some scalar field u = u(x), covarian t vector fields pLr = p?(x) an d (p,, = (pv(x),and a sym m etric tensor field gap = a,p(x).

We emphasize that for (pti = 2 = gpvpy and a,o(x) = gap(x) we obtain aparticular case of conformal maps, TV: gpLv = &'gpv (the so-called concircularmaps [15]).2.3. Invariant criterions for ng-flat spaces

DEFINITION 4. A (pseudo)-Riemannian space V is ng-flat if it admits a mapng: V-a.

We shall consider four classes of na-flat spaces denoted respectively as nCij-flatspaces, where (i) = ((0), (l), (2), (3)).It is significant that the na-maps are characterized by the corresponding invariantconditions for values similar to the Thomas parameters and the Weyl tensor (theinvariants of conformal maps [16]). Below we present the criterions for a space Vto be ng-flat.

PROPOSITION 1. For the ng-flat spaces t here are sati sfied the following cond itions:-for th e 7r(())-spaces,

W$, = R . - ;Ru.y6 ,,&, = 0; (23)-for th e 7rCl)-spaces,

3D-,Pi,$ = 2R iL p), + 6b(aP,2y, + ~u (,,S,,;-for th e T (~)-spaces,

(24)

where

R ,, = - R ,, + eF,(F,pR &, + F,pR &, - 2D,D16F,q -- 2eFiDc,F:D,FF + eFiDc,FgDrF,);

(26)

(27)


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TWISTORS AND NEARLY AUTOPARALLEL MAPS 315-for the r(+puces,Rprvys Atgu,m - gamy) + 4L7ptgarm - gawr) - vGhw45 mw~)l~ (28)

whereA=-1[~--2(~~~)1; A+B+($+ru2) =O, e=&l, (29)

for scme gradient vector p, = -$f$ nd scalar v(p) welds.We note that for the Ricci tensor of the ;?s-flat spaces from (28) and (29) it

follows the expression

2.4. The integrability conditions for q-maps equationsAil presented in this paper basic equations for the ng-maps (equations (17) (18)

and (ZO)-(22)) are systems of the first-order partial differential equations with alge-braic constraints of type (19). The integrability conditions for the ng-map equationshave been studied in [4] and, in the language of the Pffaf systems [17-201, in [5,6, 131. The most important conclusion made in the mentioned works is that we canalways verify, by using algebraic methods, whether a given system of the ng-mapequations on V is, or is not, integrable for maps to the Minkowski space. Let usillustrate this for maps nl: V -+ &J specified by the equations3(&P;-; f P;;P


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316 S. I. VACARU and S. V. OSTMintegrability conditions, we are able, in general, to clarify the question of existenceof solutions of (31) and (34).

We note that in a similar manner we can construct sets of the integrability con-ditions for the nc2)-, and r(j)-maps, cf. (20) and (22).2.5. Spinor formalism and the ng-map theory

This question was studied in detail in [13, 141 by means of deformations (byng-maps) of spin coefficients (7) in spinor covariant derivation operator (8). Usingthe g-coefficients we can transform basic ng-map equations (18), (20) and (22) andflat projectivity conditions (23), (24) and (28) into a spinor form. We omit theseconsiderations here. For our purposes it is important that for every deformation of thespin coefficients *r,:(z) (see splitting (7)) we can define a corresponding deformationtensor (see expressions (5)-(7))

(35)Putting (35), for example, into (33) we obtain a system of algebraic equations, ifnecessary in spinor variables with a spinor representation of the curvature and de-formation parameters, which permits us to answer the question whether the givendeformation of the spin coefficients generates, or not, a map rC1):V + M.

Finally, in this section, we note that every curved space V, if corresponding con-ditions on differentiability of the components of metric, connection and curvature onV are satisfied, admits a finite chain of ng-maps, i.e. a nc-transform, to the Min-kowski space M [5-9, 131. So, it is possible a new classification of the curved spacesin terms of minimal chains of ng-maps characterized by the corresponding sets ofinvariant conditions of type (23)-(25) and (28). This ng-map classification of curvedspaces differs from the well-known Petrov algebraic classification [21].3. Nearly conformal twistors

The purpose of this section is to define twistors on ng-flat spaces.3.1. Spinor equations for massless fields with spin irn (m = 0 , 1,2,. . .) and twistorequations

Let a spinor $AB...L have m indices and be symmetric:+A&L = #(A& -L). (36)

The dynamic equations for a massless spin $rn field are written asDAA~~~...~ = 0. (37)

The compatibility conditions [22, 23, l] of (37) for uncharged spinor field (36) canbe written as(m - ~)~A J~M(cT.,.K*~~~~ = 0, (38)


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TWISTORS AND NEARLY AUTOPARALLEL MAPS 317

where @LABM is the Weyl spinor on space V.Because conditions (38) are not satisfied on the generic curved spaces, there is an

essential difficulty in definition of physical fields (36) as solutions of equations (37).The same difficulty appears for twistor equations (lo), rewritten on the space V:

DjqqwB) = 0 >with the compatibility conditions

DA(CDA wB) = [email protected],A ..D

(39)

3.2. Systems of first-order partial differential equationsThe above-mentioned field and twistor equations ((37) and (10)) are systems of

the first-order partial differential equations. We shall study the general properties ofsuch systems of equations using methods of the geometrical theory of differentialequations [17-201.

Let us consider, in general form, a system of the first-order partial differentialequations on a space V, dim V/ = n :fs(1 xn,yl,..., y$$ ...) g) =o, s = 1,2, . . .) q,

where x1,. . .,x are independenttraducing new unknown variablesdyP%=@

variables and y, . . . , yT are unknown functions. In-(functions)

6 = 1)) r; & = l)... ;n,we reduce equations (40) to a Pffaf system

8 = dy _ #dxb = 0,where the variables p$ satisfy a set of finite relations:

(41)

f&a, y,&) = 0. (42)Solving (42) for q independent values p = {pi} and putting them into (41) we ob-tain a system of T Pffaf equations on i = T + nr -q unknown functions of independentvariables z? (differentials dz& play the role of distinguished variables).

Let ti be an open region locally isomorphic to Ri+n. We write the new Pffafsystem as0 = @i(z, p)d p + bf (z, p)dz = 0, (43)

where A = 1,2,. . ., r ; iL = 1,2,. . .if rank I]c$]~ 5

, T + TV_ q. Equations (42) are linearly independentr at every point x& of an open region U c V. We mention that

the integral varieties IO of the system (43) should be defined by the equations (theclosure of (43)):


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318 S. I. VACARU and S. V. OSTAF0 = 0, DOA = 0, (44)

where the quadratic exterior forms are written asDt+ = a;&dp A dp + C&&dp A dx + b/i,bdx:a A dx!

If a solutionya = yQ@) (45)

of equations (40) (or equivalently (41) and (42)) is found, it must satisfy the inte-grability conditions

dy dy-=-dyldxP dxfldya or, equivalently,

a,p; = a&p;, (46)i.e., if equations (40) are compatible, the Pffaf system (41) can be reduced to totaldifferential relations

dX = dy -p%dz* = 0.In this case solution (45) should be obtained from the relations

P(yb,xb) = c,where rank I/$$ #O. If conditions (46) are(40) by introducing new unknown functions

C = const,not satisfied, one tries to solve equation

Xjyg = pi - eXa (47)

and considering a new Pffaf system-e = dy - pzdz = 0, (48)

where yS(xir, y&,5:) = 0. To obtain a total differential relation, we multiply (48) bya nondegenerate matrix function 1_1j(x, y):

dXb = +jdyb - Cl&&&+ = 0. (49)Integrating system (49) we obtain relations-.Xb(x, y) = C, Cb = const,from which the solution y = y&(x) of equations (40) can be found in an explicitform. We note that if the deformation functions


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TWISTORS AND NEARLY AUTOPARALLEL MAPS 319where (p-)b is inverse to phi,. As particular cases we can consider the trivial inte-grating matrix, ,L$ = Sk, and (or) vanishing deformation when

or equivalently, a new Pffaf system associated to (52):-Adw -p -CCA&CC, = 0,

(52)

(53)where the unknown functions jjCfCA = *azc,, must satisfy the relations

ijA(AB) + y,,n(A.QD + I1~;A-B)ij~ = o. (54)


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320 S. . ACARU and . . OSTAFSpinors II,.,A B from (52) can be considered as those resulting from a deformationof type (47) and a multiplication by an integrating matrix as in (49). Introducing theng-maps we identify A-spinors with the deformation of spin-coefficients *$ (see

relation (7)):A...BJ'AD = dye&). (55)

PROPOSITION 2. Deformed tw istor equations (52) (and associated tw istor PfJkf sys-tern (53) and (54)) are compat ible if spinors (55) solve one of the ng-map equat ions((16), (17), (18)-(20) and (21)-(22)) and sat is& one of the corresponding ng-put cri-t erions ((23), (24), (25) and (28)).

Proof: Let us define new spin-coefficients

which according to our proposition become the trivial (with the vanishing curvature)-spin-coefficients on a flat space M. In this case equations (52) can be written asfj~'(~p) _ o-. (56)

Equations (56) are compatible because on the flat space % the Weyl tensor vanishes(see relations (11) (12)). The proposition is proved. n

Instead of the ng-maps, we can consider chains of ng-maps (nc-transforms)nc: V -+ M. Nc-twistors are defined as solutions of the deformed twistor equationswith A-spinor, being a superposition of the spin tensors

(57)associated to a finite chain of ng-maps. In a particular case when (57) reduces to(55), we obtain ng-twistors.3.4. Ng-images of twisters

On the flat space M twistors are defined as a pair of spinors, z = (gA, EA),whereWA = &),A _ izABU)~B,,- $,, = (o)~A, = const,is a general solution of the twistor equations

D -(AWB)= 0.Nc-twistors on space V, being an nc-coimage of the space M, for a given mapnc : V t iW , are defined as a pair of spinors 2 = (uA, TA,), where wA is a generalsolution of the nc-twistor equations

DA(A~B) _ AAtA.B)W~- ..D > (58)with A-spinors defined by (57). For a local common spinor coordinate system onspaces under consideration, we can write tiA = wA and define a second spinor TAl


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TWISTORS AND NEARLY AUTOPARALLEL MAPS 321as TAG = ~DAAw~. Taking into account that A, = iDAA,wA, we have

TA, 3) TA - 2 8 AA&1 . n . . .A C . (59)In a similar manner we can define the dual nc-twistors on V as the pairs ofspinors W, = (AA, p), where

AA = ()x/i + $~.;i&,~A 7 ()xA = const,and p,A = (0)pA + ixAA()XA is a general solution of the dual nc-twistor equation

DA(A~B) = npl;jJ$D.The spinors AA$~, are defined as a superposition of the ng-transformation like inequation (57).

We end this section by considering the question of the geometrical interpretationof nc-twistors. To an isotropic twistor Z = (gA, TA) # 0, ZZ, = 0 (I, = (?&, aA)denotes the complex conjugation of z) one associates [2] an isotropic line on thespace M:

xa = CO),,+ n


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322 S. . ACARU and S. . OSTAFtwistor structure, obtained as a solution of deformed twistor equations (58) withthe deformation A-spinor of type (55) will generate an ng-flat (pseudo)-Riemannianspace-time.

Firstly, we fix a spinor Ai,:o.defined as to satisfy the relations It is still not clear if the new connection $;,

where (63)p(.$q = _A,.$.$ _ X-C .ca B t E B (64)

(for simplicity we consider torsionless connections) will generate both the compati-ble ng-twistor equations (56) and the basic ng-map equations associated to a mutualtransform ng: V + M. We try to answer this question in the following way. Calcu-lating auxiliary curvature and the Ricci tensor for connection (63) and putting boththese expressions and deformation tensor (64) into relations (32)-(34), we obtain analgebraic system of equations. If this system is satisfied for some deformation par-ameters a,p and b, (see formulae (15)) it is clear that we have obtained a 7rclj-flatspace V.

PROPOSITION 3. Deformat ion spinor A.;i2,o and it s corresponding deformat ion ten-sor P&T (see (64)) generate a vacuum Einstein field if and only if it is compat iblewith the system of partial differential equations:

8, Pijy = 2(bc,.Ptiyj + awq (65)(b, and a ,0 are some covariant v ector and sym metric tensor fields, respectively ).

Proof: We sketch the proof by observing that equations (65) can be obtained bycontracting the indices Q and T in equations (18) written for a map ~(1): M, whereR aB = 0 and RorPyb = 0. Of course, to find general solutions of equations (65) in anexplicit form is also a difficult task. But we can verify, by solving algebraic equations(see Subsection 2.4), if equations (65) are, or are not, integrable. w

In a similar manner we can analyse the problem of generation of the rr(z)-flat andr(s)-flat spaces. Let us consider, for example, the 7rc2)-transforms. In this case we shallparametrize the deformation spinor (55) in such a form as to induce a deformationtensor of type (18). A-spinors should be also chosen in such a way as to induce adeformation tensor (64) satisfying conditions (25) for the r(2)-maps. Calculating anauxiliary curvature and the Ricci tensors for connection (63), and putting both tensorsinto (26) and (27) and taking into account the basic T(z)-map equations, we obtainthat criterion (25) is an algebraic equation on the tensors Rap+, Rng, F,$ and thecovariant vector field p_,.It is evident that the foregoing considerations point out at a to mutual interre-lation between integrable deformations of the twistor equations and the criterions ofinvariance and integrability of basic equations for the ng-maps rather than constitutea method to solve the Einstein equation because no explicit constructions of metrichave been considered in our study. Perhaps the twistor-gauge formulation of gravity


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TWISTORS AND NEARLY AUTOPARALLEL MAPS 323on flat nearly autoparallel backgrounds [7, 12, 141 is more convenient for the twi-stor treatment of gravity. The interrelation between nc-twistors and gauge gravity isa matter of our further investigations.

Finally, we remark that this paper contains a part of results (concerning the defi-nition of nc-twistors on locally isotropic spaces) communicated by us during the Col-loquium on Differential Geometry (Debrecen, Hungary, 25-30 July 1994) [24]. Therewe have also presented some generalizations on the spinor and twistor calculus forlocally anisotropic spaces (which generalize the Lagrange and Finsler spaces) [25, 261.The geometric constructions developed in this paper and in [12, 71 (in the frame-work of the so-called twistor-gauge treatment of gravity) admit a straightforwardextension to locally anisotropic spaces if we apply the formalism of locally aniso-tropic spinors and twistors [27, 281 and use the gauge-like formulations of locallyanisotropic gravity [26, 11, 10, 13, 141.REFERENCES

[IIPI[31[41PI161[71PIPI

[lOI11111121[I31[I41WI[I61

R. Penrose and W. Rindler: Spinors and Space-time, V.l., Cambridge University Press, Cambridge1984.R. Penrose and W. Rindler: Spinors and Space-time, V.2., Cambridge University Press, Cambridge1984.G. Vranceanu: Lectii de Geomettie differentiala, V.2., Ed. Didactica si Pedagogica, Bucuresti 1977(in Romanian).N. S. Sinyukov: Geodesic Maps of Riernannian Spaces, Nauka, Moscow 1979 (in Russian).S. Vacaru: Rom anian J. Phys. 39 (lYY4), 37.S. Vacaru and S. Ostaf: Buletinul Academiei de Stiint e a Repuhlicii Moldov a, fz ica si tehnica 3(1993) 4.S. Vacaru: Buletinul A cademiei de Stiint e a Republicii Moldov a, fizica si tehnica 3 (1993) 17.S. Vacaru and S. Ostaf: in Lagrange and Finsler Geometry, P. L. Antonelli and R. Miron (eds.).Kluwer, Dordrecht 1996, p. 241.S. Vacaru and S. Ostaf: Buletinul A cademiei de Stiinte a Republicii Moldov a, fiz ica si tehnica 1(1994) 64.S. Vacaru, S. Ostaf and Yu. Goncharenko: Rom anian .f. Phys. 39 (1994) 19Y.S. Vacaru: Contr. Int. Conf Lobachevski and Modem Geometry, Part II, V. Bajanov et al. (eds.).Kazani University Press, 1992, p. 64.S. Vacaru: tist nik Most . Univ ersity 28 (1987) 5 (in Russian).S. Vacaru: Applications of Nearly Autoparallel Maps and Tw istor-Gauge Methods in Gravit y andCondensed States, PhD Thesis. Al. I. Cuza University, Ia$, Romania 1993 (in Romanian).S. Vacaru: Tw istors and N early Aut oparallel M aps of Curved Spaces (subm. to Rep. Rom . Phys. ),K. Yano: Concircular Geom etry , I-IV . Proc. Imp. Acad. Tok yo, 16 (1940), 195; 354; 442; 505.S. A. Schouten and D. Struik: Einfihrung i n die neucren Medoden den Diff erentiol Geometrie, Bund1, 2, 1938.

[17] E. Cartan: Les Sy stem s Differentielles Exterieurs et Leurs Applicat ion Geometriques, Herman, Paris1945.

[1X] M. Haimovichi: Selected Papers, Finsler Geometry, ALL Cuza Univ., la$i, Romania lY84.[lo] S. V. Finnikov: The Method of the Cartan Exterior Differential Forms in Differential Geometry,

Gostekhizdat, Moscow 1948 (in Russian).[20] P. K. Rashevsky: Geomettic Theory of Differential Equations, Gostekhizdat, Moscow 1947 (in

Russian).[21] A. Z. Petrov: Einstein Spaces, Pergamon Press, Oxford 196Y.


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324 S. I. VACARU and S. V. OSTAF[22] H. Buchdahl: Nouvo Cim. 10 1950), 96.[23] .I. Plebadski: Actu Phys. Polon. 27 (1965) 361.[24] S. Vacaru and S. Ostaf: in Abst racts of Contributi ons t o the Colloquium on Differential Geometry, 25-30

July 1994, Lajos Kossuth University, Debrecen, Hungary 1994, p. 56.[25] R. Miron and M. Anastasiei: The Geometry of Lagrange Spaces: Theory and Applicat ions, Kluwer,Dordrecht, Boston, London 1994.[26] S. Vacaru and Yu. Goncharenko: Int. J. Theor. Phys. 34 (1995) 1955.[27] S. Vacaru: Spinor structures and nonlinear connections in vector bundles, generalized Lagrange and

Finsler spaces (to appear in J. Math. Phys. 36 (1995)).[28] S. Vacaru: Spinors in mult idimensional and locally anisot ropic spaces (in preparation).

twistors and nearly autoparallel maps

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