ON HOLLAND S FRAME FOR RANDERS SPACE AND ITS APPLICATIONS IN PHYSICS

Transcription

1 Steps n Dfferental Geometry, Proceedngs of the Colloquum on Dfferental Geometry, July, 2000, Debrecen, Hungary ON HOAND S FRAME FOR RANDERS SPACE AND ITS APPICATIONS IN PHYSICS P.. ANTONEI AND I. BUCATARU Abstract. From a rgorous Fnsler geometrc perspectve, we re-examne Holl s Rers space theory of moton of an electron n flat Mnows spacetme permeated wth an electromagnetc feld. Holl s theory was motvated by analogy wth plastc deformaton dslocaton n Bravas crystals, through wor of D. Bohm on Quantum Mechancs. We develop the anholonomc geometry of a Rers space usng Holl s frame determne two Fnsler connectons, one of them the crystallographc connecton, the other ust Cartan s connecton expressed n terms of anholonomy. Correspondng to these, we gve two fully covarant versons of Holl s theory each of whch covers the case of a curved Mnows space-tme. The crystallographc theory wth extra matter s most promsng. 0. Introducton The well-nown approach of D. Bohm (see [BH] to quantum mechancs casts quantum theory nto a classcal form so that classcal quantum physcs can be more easly compared. As a result, the so-called quantum potental arses as a gude to moton of partcles n space-tme as n de Brogle s plot-wave theory [C]. P. Holl [H 1 ], [H 2 ] noted that a draw-bac of ths quantum potental method s that t must derve from ad hoc use of Schrödnger s equaton further ased whether the converse procedure, that of reformulatng classcal theory n a manner whch s more n accord wth the sprt of quantum theory (partcularly n a way whch emphaszes the unty of feld partcle, mght not suggest an alternatve startng pont for the theoretcal treatment of quantum effects, [H 1 ]. Hs dfferental geometrc method s based on fundamental wor of S. Amar on a Fnsler approach to crystal dslocaton theory, [A]. Holl studes a unfed formalsm whch uses a anholonomc frame (nonntegrable 1-form on space-tme, a sort of plastc deformaton, arsng from consderaton of a charged partcle movng n an external electromagnetc feld n the bacground space-tme vewed as a straned medum. In fact, Ingarden [I] was frst Key words phrases. Fnsler connectons, P. Holl s Rers frame, anholonomy, plastc deformaton, electromagnetc theory. The authors would le to than Vvan Spa for her excellent typesettng of ths manuscrpt. The fst author has been partally supported by NSERC The second author was a PIMS Postdoctoral Fellow. 39

2 40 P.. ANTONEI AND I. BUCATARU to pont out that the orentz force law, n ths case, can be wrtten as a geodesc equaton on a Fnsler space called Rers space [R]. Ths results n geometrcal enttes whch depend on the electromagnetc feld (vector potental, partcle (velocty bacground space-tme parameters, [H 1 ]. The Fnsler structure mples the exstence of a global anholonomc (Holl frame whch n turn yelds a connecton wth torson vanshng Fnsler curvatures. Holl shows that the usual electromagnetc theory can be recovered from an averagng process appled to the Holl frame correspondng flat connecton, [H 1 ]. Unfortunately, Holl s theory s not truly Fnsleran as he presented t. Techncally, hs noton of covarant dfferental n Rers space (equaton (19 n [H 1 ] hs local expresson for the flat connecton (equaton (21 n [H 1 ] appear to be ncorrect. In the present paper we use Fnsler geometry to derve Holl s frame for an arbtrary Rers space; see [AIM], [M], [MR] [R] for further references on these Fnsler spaces. Holl s dea has led us to fnd Holl type frames for Kropna spaces, as well, these results wll be reported on elsewhere. The upshot of our wor so far s that t allows us to defne ntrnsc frames for C-reducble Fnsler spaces of dmenson exceedng three. Prevous worers have not succeeded n fndng parallel translaton nvarant frames because of the falure of the so-called strongly non-remannan condton, [M], [MI]. We have found these for the crystallographc connecton, defned heren. They are conformally nvarant. The fnal remars n ths paper show how to obtan two fully covarant curved space-tme versons of Holl s Theory va Theorem 3.3 or Theorem 3.4. Especally sgnfcant for the latter s the analogy wth extra matter defects n crystal lattces [GZ], [K 1 ], [K 2 ]. These may be consdered to be solutons to problems posed n [H 1 ], [H 2 ]. After prelmnary materal on Fnsler geometry n Secton 2 we ntroduce anholonomc frames, the assocated crystallographc connecton. In Secton 3 we ntroduce Holl s frame for Rers spaces prove the man Theorems of our paper. In Secton 4 we consder the orgnal set-up of Holl rederve hs results tang care to explan them n our notaton. 1. Fnsler Spaces Fnsler Connectons et M be a real n-dmensonal connected manfold of C -class (T M, π, M ts tangent bundle wth zero secton removed. Every local chart ( U, ϕ = (x on M nduces a local chart ( π 1 (U, φ = (x, y on T M. The ernel of the lnear mappng π = T T M T M s called the vertcal dstrbuton s denoted by V T M. For every u T M, Ker π,u = V u T M s spanned by { } y u. By a nonlnear connecton on T M we mean a regular n-dmensonal dstrbuton H : u T M H u T M whch s supplementary to the vertcal dstrbuton.e. (1.1 T u T M = H u T M V u T M, u T M. A bass for T u T M adapted to the drect sum (1.1 has the form ( x u = x u N (u y u, y u. The dual bass of ths s (dx, y = dy + N dx. These are

3 ON HOAND S FRAME FOR RANDERS SPACE 41 the Berwald bases. The vector feld mappng J : X(T M X(T M, defned by: (1.2 J = y dx, s globally defned on T M s called the almost tangent structure. It has the propertes: 1. J 2 = 0; 2. Im J = Ker J = V T M. See [MA] or [AIM] for more dscusson. A lnear connecton (Koszul connecton on T M s a map : (X, Y X(T M X(T M X Y X(T M for whch we have: 1. fx+gy Z = f X Z + g Y Z; 2. X fy = X(f + f X Y ; X (Y + Z = X Y + X Z. Defnton 1.1. A lnear connecton on T M s called a Fnsler connecton (or an N-lnear connecton f: 1 preserve by parallelsm the horzontal dstrbuton HT M; 2 The almost tangent structure J s absolute parallel wth respect to. For a Fnsler connecton t s mmedate that preserves also the vertcal dstrbuton. In the Berwald bass ( x, y adapted to the decomposton (1.1, a Fnsler connecton can be expressed as: x x = F x ; x y = F y (1.3 y x = C x ; y y = C y. Observe that under a change of nduced coordnates on T M the functons F transform le the coeffcents of a lnear connecton on M C as the components of a (1, 2 tensor feld on M. We wll say that C s a (1, 2-type Fnsler tensor feld. In general, a tensor feld of (r, s-type on T M s called Fnsler tensor feld (or d-tensor feld f under a change of nduced coordnates on T M ts components transform le the components of a (r, s type tensor on the base manfold M. For X = X x, a horzontal vector feld, the absolute dfferental wth respect to the Fnsler connecton s gven by: (1.4 X = dx + F X dx + C X y = dx + ω X, where ω = F dx + C y s the connecton 1-form of. The formula (1.4 can be wrtten n an equvalent form: (1.4 X = X dx + X y.

4 42 P.. ANTONEI AND I. BUCATARU Here, X X are the horzontal the vertcal covarant dervatves of X, respectvely. These must satsfy (1.5 X x x = X x ; X y x = X x,.e. (1.5 X = X x + F rx r ; X = X y + C rx r. Of course, the horzontal vertcal covarant dervatves wth respect to a Fnsler connecton, can be defned n general for a Fnsler tensor feld. For example, f T are the components of an (1, 1 Fnsler tensor feld then h v-covarant dervatves are gven by: (1.6 T = T x + F rt r F r T r T = T y + C rt r C r T r. For a Fnsler connecton one consders typcally T (X, Y = X Y Y X [X, Y ] R(X, Y Z = X Y Z Y X Z [X,Y ] Z the torson the curvature. It s well nown [AIM], [MA] that n the bass ( x, y there are only fve components of torson three components of curvature. The fve components of torson are: (1.7 ( ht x, x =: T x = (F F x ; ( ( vt x, x =: R y = N x N x ( ht y, x = C x ; ( ( vt y, x =: P N y = y F ( vt y, y =: S y = ( C C y y ; y ; (h h-torson (v h-torson (h hv-torson (v hv-torson (v v-torson.

5 ON HOAND S FRAME FOR RANDERS SPACE 43 The three components of curvature are gven by: Rh = F x h F h x + F F m mh FhF m m + CmR h m (1.8 P h = F y h S r = C y r C h + C mp m h C r y + Cm C mr C m rc m For a Fnsler connecton we have the followng Rcc dentty X X = Xm R r r mr X T m r m X m Rr m (1.9 X r X r = X m Pmr X Cm m r X m Pr m X r X r = X m Smr X m Sr. m 2. The Parallel Dsplacement Determned by an Anholonomc Fnsler Frame et V be an open subset on T M (2.1 Y : u V Y (u V u T M, = 1, n be a vertcal frame over V. Assume that V = T M. If we consder Y (u = Y(u y u, then ( Y(u are the entres of a nonsngular matrx. Denote by (Y the components of the nverse of these matrx. Ths means that: (2.2 Y Y =, Y Y β = β. We call (Y an anholonomc Fnsler frame. Naturally, every geometrc obect feld on T M can be expressed n anholonomc Fnsler frames. For example, f T are the components of a (1, 1 Fnsler feld, then the anholonomc components are gven by: (2.3 T β = Y T Y β. Consder the anholonomc Berwald bass: (2.4 x = Y x y = Y y. et be a Fnsler connecton wth the holonomc coeffcents (F, C gven by (1.3. As we have seen preserves, by parallelsm, the horzontal vertcal dstrbuton, so n the anholonomc Berwald bass ( x, y the Fnsler connecton

6 44 P.. ANTONEI AND I. BUCATARU can be expressed as: (2.5 x y x β = F γ β x β = Cγ β x γ, x x γ, y y β = F γ β y γ y β = Cγ β y γ. The functons (F γ β, Cγ β are called the anholonomc coeffcents of the Fnsler connectons,. Proposton 2.1. et be a Fnsler connecton wth the holonomc coeffcents (F, C (Y be an anholonomc Fnsler frame. Then, the anholonomc coeffcents of the Fnsler connecton are gven by: (2.6 (2.6 F γ β = ( Y x = Y Y βy γ C γ β = ( Y y = Y Y βy γ + Y F Y βy γ = Y γ Y Y β + Y C Y βy γ = Y γ Y Y β. = Y ( Y γ x ( = Y Y γ y + F Y γ + C Y γ Conversely, f for a Fnsler connecton we now the anholonomc coeffcents (Fβγ, C βγ then the holonomc coeffcents are gven by: ( Y F = Y Y β Y β = Y + Y Fβ Y β = Y β Y Y β = x β ( Y x β + Y ( Y C = Y Y β Y β = Y = Y β Y Y β = F β Y β + Y Cβ ( Y Y β + Y Cβ Y Y β Y β Y Y β. Proof. The frst formulae for F γ β Cγ β can be obtaned f we compare (1.3 (2.5. For the second we have to tae nto account Y Y x β Y Y y β = Y Y x β ; = Y Y y β ; whch follows from (2.2 by dfferentaton. Y Y Y β x Y β y = Y β Y x = Y β Y y Y β Y β

7 ON HOAND S FRAME FOR RANDERS SPACE 45 We then fnd that for any Fnsler connecton (1.3 [ ] (a x, x β = Ω γ β x γ + Rγ β y γ (2.7 (b [ ] x, y β = (P γ β + F γ β y γ + Ωγ (β x γ (c [ ] y, y β = η γ β y γ. See [MI] for more dscusson. Here, R γ β are the anholonomc components of the (1, 2 type tensor feld R gven by (1.7, Ωγ β Ωγ (β are the anholonomc obects defned by ( Y (a Ω γ β = Y γ β x Y x β (2.8 (b Ω γ (β = Y Also, the (v-hv-torson has components (2.8 (c P γ β = ( N y whle the (v-v-torson s (2.8 (d η γ β = Y γ ( y Y γ Y β. F Y γ Y Y β ( Y β y Y y β The frame Y s holonomc f only f there exsts n functons ϕ on M for whch Y = ϕ x,.e. the 1-form Y vdx s exact. Proposton 2.2. The frame Y s holonomc f only f Ω γ β = 0 = Ωγ (β. Proof. Ω γ (β = 0 mples Y s ndependent of y, then Ω γ β = 0 mples Y are gven by gradents of n functons on M. Remar. Y s holonomc f only f [ x, x β ] [ x, y β ] are n V T M,.e. ff these e bracets are vertcal. Of course, 2.7(c expresses the fact that the fbre of T M s ntegrable. et X be the anholonomc components of a Fnsler vector feld X. Then the absolute dfferental of X wth respect to the Fnsler connecton can be expressed as: (2.9 X = dx + F βγx β dx γ + C βγx β y γ. (Here, the reader may note the dfference wth (19 n [H 1 ]..

8 46 P.. ANTONEI AND I. BUCATARU Proposton 2.3. Gven an anholonomc Fnsler frame (Y, there exsts a unque Fnsler connecton for whch the gven frame s h- v-covarant constant. We call ths the Crystallographc connecton. Proof. From (2.6 we have that Y = 0 s equvalent wth F γ β = 0 (2.10 F = Y Y x Smlarly, Y = 0 C γ β (2.10 C = Y Y y = Y γ x Y γ. = 0 whch s also equvalent wth: = Y γ y Y γ. (The reader may note the dfference between (2.10 (21 n [H 1 ]. Proposton 2.4. For the Fnsler connecton gven by Proposton (2.3, all three components of curvature vansh. Proof. Accordng to Y = 0, Y = 0 the Rcc denttes (1.9 we obtan Rl = P l = S l = The Anholonomc Fnsler Frame Determned by a Rers Metrc (The Holl Frame et us consder a Fnsler space F n = (M, on an n-dmensonal manfold M. Ths means that: 1 : T M R s of C -class contnuous on the zero secton; 2 s postvely homogeneous wth respect to y; 3 The matrx wth the entres: (3.1 a = y y has a constant ran n on T M. It s nown that a Fnsler space has a canoncal nonlnear connecton HT M, wth the local coeffcents: (3.2 N = 1 2 y (γ ly y l, where γl are the Chrstoffel symbols of second nd for the metrc tensor a. There s also a Fnsler connecton (Cartan connecton wth the holonomc coeffcents gven by: (3.3 F = 1 2 ar ( ar x C = 1 2 ar ( ar y + a r x + a r y a x r a y r As s well-nown, ths connecton s metrcal (a = 0 a = 0 h v symmetrc (T = 0 S = 0 [AIM], [MA]..

9 ON HOAND S FRAME FOR RANDERS SPACE 47 Together wth the Fnsler space F n = (M, we shall consder a covector feld b (xdx on M (or an open set V of M. Then β(x, y = b (xy s a scalar functon on T M (or on π 1 (V. The functon : T M R, defned by: (3.4 (x, y = (x, y + β(x, y s also the fundamental functon of a Fnsler space [AIM], [MR]. The par (M, s called a Rers space. Denote by: (3.5 g = y y the fundamental tensor of the Rers space (M,. Tang nto account the homogenety of we have: p := 1 y = a y ; p := a p = y l := 1 (3.6 y = g y ; l := g l = y = p + b l = p ; l l = p p = 1; l p = ; p l = b p = β, b l = β. The metrc tensors (a (g are related by: (3.7 g = a + b p + p b + b b β p p = (a p p + l l. et us consder now ( X (x, y an arbtrary, but fxed anholonomc frame (but t could be also holonomc. Denote by a β (x, y = X (x, yx β (x, ya (x, y, the components of the metrc (a wth respect to (X. If (X are the components of the nverse matrx of (X, denote: Consder (Y l = X l, p = X p, l = X l p = X p. the matrx wth the entres: Y (x, y = ( l l + p p. Then ths matrx s nvertble the components of ts nverse are: ( (Y 1 = + l l p p.

10 48 P.. ANTONEI AND I. BUCATARU Theorem 3.1. For a Rers space (M, consder n the case > 0 : ( (3.8 Yγ = γ l l γ + p p γ defned on open set V n T M where > 0. Then (Y γ = Yγ anholonomc Fnsler frame. y γ=1,n s an We call t, Holl s frame of Rers space [H 1 ], [H 2 ]. It s p-homogeneous of degree zero n y a conformal nvarant n the sense that e φ(x leaves Yγ fxed. Proof. Consder also: (3.9 Y γ = γ Y γ ( γ + lγ l p γ p. We have to chec that Y = Y ( ( Yγ Y γ = γ + l l γ p p γ γ lγ l + = l l + p p + l l p p + p p γ l γ l p p =. γ Y β = γ β. et us verfy the former: pγ p l l + l l γ p γ p Theorem 3.2. Wth respect to Holl s frame the holonomc components of the Fnsler metrc tensor (a β s the Rers metrc (g, that s: (3.10 g = Y Y β a β. Proof. We have Y β a β = ( β + lβ l p β p a β = a + p l p p ( ( Y Y β a β = + l l p p a + p l = a + p l p p + p l + l l l p p p p p p l + p p

11 ON HOAND S FRAME FOR RANDERS SPACE 49 = (a p p + l l = g. Corollary. We have Y p = l, Y p = l. Theorem 3.3. Consder the Rers space (M, Holl s Frame (Y. Then there exsts a unque Fnsler connecton (Cartan connecton wth local coeffcents (Fβγ, C βγ for whch (1 a β γ = 0 a β γ = 0 (2 the anholonomc components of the (h h-torson (v v-torson are τ βγ := F γβ F βγ = Ω βγ Σ βγ := C γβ C βγ = η βγ. Proof. et us consder the Cartan connecton of the Rers space (M,. Denote by (F, C t s holonomc coeffcents. Ths s the unque Fnsler connecton whch satsfes Matsumoto s Axoms: (a g = 0 g = 0; (b T = 0 S = 0. Consder (Fβγ, C βγ the anholonomc coeffcents of ths Cartan connecton (2.6. All we have to prove s that (1 (a are equvalent also (2 (b. Frst we prove that (3.12 g = Y Y β a β γy γ g = Y Y β a β γ Y γ. et us start wth the (RHS of (3.12 1, ( a β γ Y Y β Y γ = aβ x γ a βfγ a Fβγ Y Y β Y γ = Y γ a β x γ Y Y β a βy β F γy γ Y a Y FβγY γ Y β = a β x Y Y β g ly l FγY γ = x (g lmy l Y m β g l ( Y l β x + Y m β Y Y β F l m Y g l Y β g l Y l F βγy γ ( Y l x + Y m F l m Y β Y = g x g lf l g l F l Y l + g l x Y Yβ m + g m x Y β

12 50 P.. ANTONEI AND I. BUCATARU We have used: a β Y β Y l g l x Y Yβ l g l x Y β = g. = g l Y l a Y = g l Y l accordng to (3.10 FβY l Y β = Y l x + Y m Fm l accordng to (2.6. A smlar formula wors for v-covarant dervatve. We also have, as n [MI], τβγ + Ω βγ = TY Y β Y γ C βγ + Ω β(γ = C Y Y β Y γ (equvalent to (2.6 2 βγ +η βγ = S Y Y β Y γ. Usng these formulae we have that (2 (b are equvalent. Remar. From the second formula of (3.13 we can determne the vertcal anholonomc coeffcents of the Cartan connecton as: C βγ = Ω β(γ + C Y Y β Y γ, where C = 1 4 gl 3 2 y l y y. Theorem 3.4. Consder the Rers space (M, wth Holl s Frame. The crystallographc connecton satsfes: 1 g = Y Y β g = Y Y β Y γ a β Y γ x γ a β y γ. 2 The holonomc components of the (h h-torson, (v v-torson (h hvtorson are: T = Ω βγy Y β Y γ ; S = η βγy Y β Y γ C = Ω β(γ Y Y β Y γ, respectvely. 3 Ths connecton s flat Y = 0, Y = 0. Proof. Accordng to Propostons , the anholonomc components for the crystallographc connecton are Fβγ = C βγ = 0. Then a β γ = a β x γ a β γ = a β y. As (3.12 holds also for ths connecton we have that (1 s verfed. γ Snce Fβγ = C βγ = 0 then, βγ = τ βγ = 0 (3.13 gves (2. Proposton 2.4 gves the condton (3. Remar. Snce the Holl frame s a conformal nvarant t follows from (2.8 b (2.8 d that the anholonomc obects Ω β(γ η βγ are conformal nvarants.

13 ON HOAND S FRAME FOR RANDERS SPACE 51 Also from Theorem 3.4, 2 we have that the (v v-torson S (h hv-torson of the crystallographc connecton are conformal nvarant. C 4. Anholonomc Geometry of Flat Remannan Space Versus Rers Space et (a be a regular Remannan metrc possbly wth non-postve-defnte sgnature. Suppose (M, a s a flat space. Denote by (γ the Chrstoffel symbols of second nd of the Remannan metrc. Snce the Remannan space (M, a s flat there exsts a frame X(x (holonomc or not on the base manfold such that a β = X(xX β (xa (x are constants. Wth respect to ths frame the coeffcents of ev-cvta connecton are γβγ = 0. Then (x, y = a (xy y = a β y y β s the fundamental functon of a Fnsler metrc. If we assume that the manfold M s endowed wth a covector feld b = b (xdx, then : T M R, gven by (x, y = a (xy y + b (xy s the fundamental functon of a Rers space. Theorem 4.1. For the Rers space (M, above wth Holl s Frame, the Cartan connecton s the unque connecton whch n anholonomc coordnates satsfes: 1 a β γ = 0 a β γ = 0; 2 τ βγ = Ω βγ βγ = η βγ. The anholonomc coeffcents of the Cartan connecton are gven by (4.1 F βγ = 1 2 a (a βε Ω ε γ + a γε Ω ε β a ε Ω ε βγ C βγ = 1 2 a (a βε η ε γ + a γε η ε β a ε η ε βγ. Proof. Accordng to Theorem 3.3 the Cartan connecton s the unque Fnsler connecton of the Rers space (M, for whch (1 (2 hold. From (1 (2 by some stard computaton we can get: F βγ = 1 2 a ( aβ x γ C βγ = 1 2 a ( aβ y γ + γ x β + a γ y β a βγ x a (a βε Ω ε γ + γε Ω ε β a ε Ω ε βγ a βγ y a (a βε ηγ ε + a γε ηβ ε a ε ηβγ. ε Tang nto account that (a β are constants we get (4.1. Denote now by s the arc length wth respect the Remannan metrc,.e. ds = (a (xdx dx 1/2 = (a β dx dx β 1/2 by S the arc length wth respect

14 52 P.. ANTONEI AND I. BUCATARU to Fnsler metrc,.e. (4.2 ds = (a (xdx dx 1/2 + b (xdx = (g (x, ydx dx 1/2. It s nown that f we perform a varaton of (4.2 we get the orentz force equaton: d 2 x (4.3 ds 2 + dx dx γ ds ds = F dx ds, where γ are the Chrstoffel symbols of the Remannan metrc (a F = a ( b x b s the electro-magnetc tensor feld. But the equaton (4.3 has an x equvalent form: (4.3 d 2 x ds 2 + F dx dx ds ds = 0, where F are coeffcents of the Cartan connecton of the Rers space (M,. In anholonomc coordnates gven by Holl s Frame, the equaton (4.3 becomes: d 2 x (4.4 ds 2 + F βγ dx β dx γ ds ds = 0. If we tae nto account the Theorem (4.1, then the orentz equaton (4.3 can be expressed anholonomcally as: (4.5 d 2 x ds 2 + a (a βε Ω ε γ dxβ ds dx γ ds = 0. Theorem 4.2. et (M, be a Rers space as above wth Holl s Frame crystallographc connecton. Then τβγ = 0, βγ = 0 C βγ = 0. Moreover, ths Fnsler connecton s flat metrc Y = 0, Y = 0. Proof. As (a β are constant we have by Theorem 3.4(1 that the crystallographc connecton s metrc. Flatness follows from Proposton 2.4 the parallel translaton nvarance of Y from Proposton 2.3. Remar. As for the crystallographc connecton, we have the anholonomc coeffcents Fβγ = C βγ = 0, the nduced absolute dfferental s: (4.6 X = dx. Consequently, the geodesc equatons for ths connecton are d2 x ds 2 = 0. If the Remannan metrc s postve defnte we can choose the frame ( X (x such that X (xx β (xa (x = β. In ths case accordng wth (3.10 we have: ths means exactly that ( Y = Y g Y Y β = β y =1,n s an orthogonal frame.

15 ON HOAND S FRAME FOR RANDERS SPACE 53 Fnal Remar. Consder the absolute dfferental wth holonomc coeffcents (2.9, (2.10 (2.10 n the present paper compare them to (19 (21 n [H 1 ]. For the case of Theorem 4.2 these two sets are dentcal. However, ntroducton of matter curves the flat Mnows s 4-space of specal relatvty. Holl clams that a fully covarant theory (.e. a curved space verson of Theorem 4.2 s stll possble. Theorem 3.3 s one way to complete hs program. In ths case the Cartan torson tensor plays a crucal role the three curvature tensors wll not vansh. Unfortunately, the Holl frame s not nvarant under parallel translaton. Theorem 3.4 also completes Holl s program. Indeed, not only s Holl s frame nvarant, but equaton (1 n the statement of Theorem 3.4 has the character of extra matter caused by pont defects [K 2 ]. Pont defects n Bravas crystals are accounted for by non-metrcty. Thus, matter n Mnows 4-space s expressed accordng to defect theory va (1. It s obvous that ths descrpton s n the sprt of Holl s orgnal dea, based on the geometry of defects n crystal lattces, [H 1 ], [H 2 ]. References [A] Amar, S., A theory of deformatons stresses of Ferro magnetc substances by Fnsler geometry, RAAG Memors of the Unfyng Study of Basc Problems n Engneerng Physcal Scences by Means of Geometry (K. Kondo, ed., vol 3, Gauutsu Bunen, Fuyu- Ka, Toyo, 1962, pp [AIM] Antonell, P.., Ingarden, R.S. Matsumoto, M., The Theory of Sprays Fnsler Spaces wth Applcatons n Physcs Bology, Kluwer Academc Press, Dordrecht 1993, pp [BH] Bohm, D. Hley, B.J., The Undvded Unverse, Routledge, ondon New Yor [C] Cushng, James, T., Quantum Mechancs, Unv. of Chcago Press [GZ] Günther, H. Zoraws, M., On the geometry of pont defects dslocatons, Ann. der Phys. 7, (1985, [H 1 ] Holl, P.R., Electromagnetsm, partcles anholonomy, Phys. ett. 91, (1982, [H 2 ] Holl, P.R. Phlppds, C., Anholonomc deformatons n the ether: a sgnfcance for the electrodynamc potentals, Quantum Implcatons, (B.J. Hley F.D. Peat, eds., Routledge Kegan Paul, ondon New Yor, 1987, pp [I] Ingarden, R.S., Dfferental geometry physcs, Tensor, N.S. 30, (1976, [K 1 ] Kröner, E., The contnuzed crystal a brdge between mcro- macromechancs, Z. angew. Math. Mech. 66, (1986, 5, [K 2 ] Kröner, E., The dfferental geometry of elementary pont lne defects n Bravas crystals, Int. J. Theor. Phys. 29, (1990, [M] Matsumoto, M., Theory of Fnsler Spaces wth (, β-metrc, Rep. Math. Phys. 31, (1992, [MA] Mron, R. Anastase, M., The Geometry of agrange Spaces: Theory Applcatons, Kluwer Academc Press, Dordrecht, [MI] Mron, R. Izum, H., Invarant frame n generalzed metrc spaces, t Tensor, N.S. 42, (1985, [MR] Mron, R., General Rers spaces, n agrange Fnsler Geometry. Applcaton to Physcs Bology, (P. Antonell R. Mron, eds., Kluwer Academc Press, Dordrecht (1996,

16 54 P.. ANTONEI AND I. BUCATARU [R] Rers, G., On an asymmetrc metrc n the four-spaces of general relatvty, Phys. Rev. 59, (1941, Department of Mathematcal Scences, Unversty of Alberta, Edmonton, Alberta, Canada, T6G 2G1. E-mal address: pa2@gpu.srv.ualberta.ca Faculty of Mathematcs, Al.I.Cuza Unversty, Iaş, Iaş 6600, Romana E-mal address: bucataru@uac.ro