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2 CARPATHIA J. MATH , o. 2, Onlne verson avalale at Prnt Edton: ISS Onlne Edton: ISS Metrzale systems of autonomous second order dfferental equatons M. CRASMAREAU ABSTRACT. Well-known notons from tangent undle geometry, namely nonlnear connectons and semsprays, are extended to undle-type tangent manfolds. As man suject, the metrzalty of oth semsprays and nonlnear connectons s nvestgated through Oata operators. 1. ITRODUCTIO Almost tangent structures were ntroduced y Clark and Bruckhemer [4] and Elopoulos [10], [11], [12] around 1960 and have een nvestgated y several authors, n [1], [5], [9], [20]. As s well-known, the tangent undle of a manfold carres a canoncal ntegrale almost tangent structure, hence the name. Ths almost tangent structure play an mportant role n the Lagrangan descrpton of analytcal mechancs [9], [13], [17]. The am of present paper s to extend two natural ojects, namely nonlnear connectons and semsprays, from tangent undles to tangent manfolds geometry. The frst geometrcal oject s studed y means of vertcal projectors and the second mples the exstence of a gloal vector feld of Louvlle type. The paper s structured as follows. In the second secton nonlnear connectons are ntroduced and nterpreted as kernels of vertcal projectors and the equvalence wth other two types of vector 1-forms s proved. In the thrd secton the noton of second order dfferental system semspray on short s defned and the relatonshp etween semsprays and nonlnear connectons s dscussed n detal. As partcular case, the noton of spray corresponds to a homogenety condton. Types of curves assocated n a natural manner to nonlnear connectons and semsprays are studed n the next secton. The sgnfcance of the noton of semspray from the pont of vew of dfferental equatons s the theme of secton 4. The man part of the paper, namely the secton 5, s devoted to the metrzalty prolem of oth semsprays and nonlnear connectons. Let us remark that prevously, the metrzalty of nonlnear connectons on tangent undles was studed y Bucataru n [3]. Therefore, our theorem 5.1 s a generalzaton of Theorem 2.4 of [3]. Other very nterestng studes on the metrzalty prolem are provded y Olga Krupkova n [14] and [15] n the usual framework of tangent undles. Thus, our noton of metrc on a tangent undle gves a natural generalzaton and can e put n correspondence wth the man noton of paper [16]. Receved: ; In revsed form: ; Accepted: Mathematcs Suject Classfcaton. 58A30; 34A26; 37C10; 53C15. Key words and phrases. Bundle-type tangent manfold, vertcal projector, semspray, metrzalty. 163

3 164 M. Crasmareanu 2. OLIEAR COECTIOS O TAGET MAIFOLDS Let M e a smooth, m-dmensonal real manfold for whch we denote: C M-the real algera of smooth real functons on M, X M-the Le algera of vector felds on M, T r s M-the C M-module of tensor felds of r, s-type on M. An element of T 1 1 M s usually called vector 1-form [18, p. 176]. The framework of our paper s fxed y: Defnton 2.1. J T 1 1 M s called almost tangent structure on M f: 2.1 mj = ker J. The par M, J s an almost tangent manfold. The name s motvated y the fact that 2.1 mply the nlpotence J 2 = 0 exactly as the natural tangent structure of tangent undles, [17]. Denotng rankj = n t results m = 2n. In addton, we suppose that J s ntegrale.e.: 2.2 J X, Y := [JX, JY ] J [JX, Y ] J [X, JY ] + J 2 [X, Y ] = 0 and n ths case J s called tangent structure and M, J s called tangent manfold. In the followng we shall work only on tangent manfolds. From [19] we get: the dstruton mj = ker J defnes a folaton denoted V M and called the vertcal dstruton. Example 2.1. M = R 2, J x, y = 0, x s a tangent structure wth ker J the Y-axs, hence the name. there exsts an atlas on M wth local coordnates x, y = x, y 1 n such that J = y dx.e.: 2.3 J x = y, J y = 0. We call canoncal coordnates the aove x, y and the change of canoncal coordnates x, y x, ỹ s gven y [19]: { x = x x 2.4 ỹ = x x y a + B x. a It results an alternatve descrpton n terms of G-structures. amely, a tangent structure s a G-structure wth: { } A On G = C = GL2n, R; A GLn, R B A On O and G s the nvarance group of the matrx J = n.e. C G f and I n O n only f C J = J C. Inspred y Defnton 1.1 of [2, p. 71] we gve a frst man noton: Defnton 2.2. A vector 1-form v : X M X M satsfyng: { J v = v J = J

4 s called vertcal projector. Metrzale SODE 165 From mv ker J = V M and from v mj = 1 V M. In concluson mv = V M and v 2 = v; these facts explan the name of v. Another well-known noton n tangent undles geometry extends to: Defnton 2.3. [19] A supplementary dstruton to the vertcal dstruton V M: 2.6 X M = V M s called normalzaton or horzontal dstruton or nonlnear connecton. A vector feld elongng to s called horzontal and one elongng to V M s called vertcal. Because a vertcal projector v s C M-lnear wth mv = V M we have a frst mportant result: Proposton 2.1. A vertcal projector v yelds a nonlnear connecton denoted v through relaton v = ker v. Ths relaton s a generalzaton of remarks from [2, p. 71] where the tangent undles case s treated. An mportant remark s that the last result admts a converse. amely, f s a nonlnear connecton let h, v the horzontal and vertcal projecton wth respect to the decomposton 2.6. Proposton 2.2. v s a vertcal projector wth v =. Proof. From mv = V M = ker J t follows v eng projector satsfy v V M = V M = mj and then we have The second fact s mmedately from the defnton of v. Wth respect to the dentfcaton nonlnear connecton=vertcal projector let us pont other two equvalent choces: I Followng [13] we get: Defnton 2.4. A vector 1-form Γ s called nonlnear connecton of almost product type f: { Γ J = J 2.7 J Γ = J. Proposton 2.3. If Γ s a nonlnear connecton of almost product type then: v Γ = 1 2 1X M Γ s a vertcal projector, V M s the 1-egenspace of Γ, v Γ s the +1-egenspace of Γ. It results that every vertcal projector v yelds a nonlnear connecton of almost product type: Γ = 1 X M 2v. From ths last relaton t results Γ 2 = 1 X M.e. Γ s an almost product structure on M hence the name. Proof. J v Γ = J J Γ = J J = 0 and v Γ J = J Γ J = 2 1 J + J = J. 2

5 166 M. Crasmareanu V M = mv Γ = {X X M ; Γ X = X}. v Γ = ker v Γ = {X X M ; Γ X = X}. II Inspred y [18, p. 180] we defne: Defnton 2.5. A vector 1-form h s called horzontal projector f: { h = h ker h = V M. Proposton 2.4. If h s a horzontal projector then: v h = 1 X M h s a vertcal projector, v h s the +1-egenspace of h. It follows that every vertcal projector v yelds a horzontal projector: h = 1 X M v. Proof. From h 1 X M h = 0 we have m 1 X M h ker h = V M = ker J then J v h = 0. Also, mj = V M = ker h mply v h J = J h J = J. v h = ker v h = {X X M ; h X = X}. In canoncal coordnates a vertcal projector reads: 2.9 v = j y dxj + y dy = y jdx j + dy and the functons j x, y are called the coeffcents of v respectvely 1,j n { δ v. A ass of X M adapted to the decomposton 2.6 s δx := x } j y j, y called Berwald ass. Then: v = 1 n y δy, h = δ δx dx where {dx, δy = dy + j dxj } s the dual of Berwald ass. 3. SEMISPRAYS O BUDLE-TYPE TAGET MAIFOLDS In the followng we suppose that V M admts a gloal secton E = y y called Euler vector feld after [19] on tangent undles E s called Louvlle vector feld, [2, p. 70]. Agan after [19] the trple M, J, E wll e called undle-type tangent manfold and n ths case B from are zero cf. [19]. For examples of undle-type tangent manfolds see [19]. As n the tangent undle case [2, p. 70] we gve a second man noton: Defnton 3.6. If M, J, E s a undle-type tangent manfold then S X M s called semspray or second order dfferental equaton sode on short f: 3.10 J S = E. In canoncal coordnates: 3.11 S = y x 2G x, y y and the functons G x, y are the coeffcents of S. Another mportant result s:

6 Metrzale SODE 167 Proposton 3.5. A vertcal projector v yelds an unque horzontal semspray denoted S v. Proof. Ths proposton s a generalzaton of a smlar result wthout proof from [2, p. 71]. The formula: 3.12 G = 1 2 jy j gves the concluson. In other words: 3.13 S v = y δ δx. The converse of last result s: Proposton 3.6. If S s a semspray then v S : X M X M gven y: 3.14 v S X = 1 X + [S, JX] + J [X, S] 2 s a vertcal projector. Proof. Because: J v S X = 1 JX J [JX, S], 2 v t must to prove that: 3.15 J [JX, S] = JX for every X X M. But from 2.2 wth Y = S: 3.16 [JX, E] J [JX, S] J [X, E] = 0 and then 3.15 s equvalent wth: S J X = 1 JX + J [JX, S] [JX, E] = J [X, E] + X. Case 1 X = [ x ] ya y, y a = y = J x.e s true for ths case. Case 2 X = [0, E] = 0 = J y y + y.e s true agan. If S s gven y 3.11 then the coeffcents of v S are: 3.18 j = G y j. A frst natural queston s: gven the vertcal projector v = j there exsts a semspray S such that v = v S? Lookng at 3.18 t results that j must e a gradent wth respect to y. Then f we defne: t k j = k y j k j y t results: Corollary 3.1. There exsts a semspray S such that v = v S f and only f t k j = 0, 1, j, k n.

7 168 M. Crasmareanu A second natural queston s wth respect to the sequence: when S = S v S? S v S S v S G 3.18 G 3.12 y j 1 G ; 2 y j yj Corollary 3.2. Let S e a semspray and v S the assocated vertcal projector. Then S s exactly S v S gven y Proposton 3.5 f and only f: 3.19 [E, S] = S. Proof. v S S = S + [S, E] = 0. Defnton 3.7. A semspray satsfyng 3.19 wll e called spray. Locally 3.19 means: G = y j G y j.e. the functons G are homogeneous of degree 2 wth respect to varales y. In terms of the assocated vertcal projector v S = j t results, usng 3.18, that j are homogeneous of degree 1 wth respect to y : 3.21 j = y a j y a. The aove formulae can e put n a compact form usng the Frölcher-jenhus formalsm. Recall that for a vector 1-form K and Z X M we have the racket [K, Z] F : X M X M gven y [18, p. 177]: 3.22 [K, Z] F X = [K X, Z] K [X, Z] where n the R.H.S. we have the usual Le racket of vector felds. Then 3.14 ecomes: 3.23 v S = 1 1X M [J, S] 2 F and lookng to Proposton 2.3 t results that [J, S] F s exactly the nonlnear connecton of almost product type Γ assocated to v S. Corollary 3.3. A semspray S s a spray f and only f: 3.24 [v S, E] F = 0. Proof. Let X X M. The aove relaton means: [v S X, E] = v S [X, E]. I X = [ ] [ ] y y, E = v S y, E whch s true ecause [ ] y, E = E V M, II X = δ [ ] δ δx 0 = v S δx, E = v S y a j y a j y j = y a j y a j whch s equvalent wth characterzaton yj

8 Metrzale SODE 169 A thrd natural queston s wth respect to the sequence: v S v v Sv j 3.12 G = 1 2 k 3.18 yk G y j ; when v = v Sv? We must have: j = 1 2y j k y k = 1 2 j yk k and then: yj Corollary 3.4. Let v = j e a vertcal projector and S v the assocated semspray. Then v s exactly v Sv gven y Proposton 3.6 f and only f: 3.25 j = y k k y j. If v = v Sv then S v s a spray and then t k j = 0 and 3.21 holds. A last queston s: gven the semspray S = G there exsts a vertcal projector v such that S = S v? So, we must to solve the system G = j yj n the unknowns j. We don t know the general answer, yet, ut s ovously that f S s spray then the answer s postve wth v = v S. 4. PATHS OF OLIEAR COECTIOS AD SEMISPRAYS Let e a nonlnear connecton wth assocated vertcal projector v = j. { } δ Wth respect to the Berwald ass δx, y we have: 1 n [ ] δ δ δx, δx j = R a j y [ ] a δ 4.26 δx, y j = a y j y [ ] a y, y j = 0 where: 4.27 Rj a = δ a δx j δ j a δx. Then the horzontal dstruton s ntegrale f and only f: Rj k = 0, 1, j, k n. Let us suppose that v = v S for the semspray S [ = G. From 3.23 we get that X s symmetry for v S f and only f: 1X M [J, S] F, X ] = 0; ut [ F 1X M, X ] = 0 for every X and then X s symmetry for v F S f and only f: 4.28 [[J, S] F, X] F = 0. Lookng at local expressons let us note that Rj a for v S s: 4.29 Rj a = δ G a δx j y δ G a δx y j.

9 170 M. Crasmareanu Snce we are nterested n dynamcs let us study curves on undle-type tangent manfolds. Let c = c t e a curve on M wth local expresson c t = x t, y t = x t, y t. Three cases are of mportance: I c s an ntegral curve of the semspray S. It results from 3.11 the dfferental system: dx dt t = y t dy 4.30 dt t + 2G x t, y t = 0 whch explans the name sode for S. II the tangent feld of c s horzontal wth respect to the vertcal projector v. From 2.9: dc dx 4.31 v = v dt dt x + dy dt y = j dx j dt + dy dt y. Such a curve s called h-path of v and s soluton of dfferental system: 4.32 dy dt t + j x t, y t dxj t = 0. dt III a h-path of v satsfyng n addton dx dt = y wll e called h-ntegral curve of v and s soluton for: dx dt t = y t 4.33 dy dt t + j x t, dx dx j. dt dt t = 0 Wth respect to Proposton 3.5 comparng 4.30 and 4.33 t results va 3.12: Proposton 4.7. A h-ntegral curve of v s an ntegral curve of S v. Wth respect to Proposton 3.6 there s no relaton etween ntegral curves of S and v S n the general case. But n the homogeneous case we get: Proposton 4.8. If S s a spray then an ntegral curve of S s a h-ntegral curve of v S. 5. THE METRIZABILITY PROBLEM 5.1. The general prolem of metrc pars. Let us fx a semspray S = G and a nonlnear connecton = j. Recall that, after 3.18, S produces a nonlnear connecton = c c j= G y j, c from canonc. Followng [3] let us consder: Defnton 5.8. The dynamcal dervatve assocated to the par S, s the map S : V M V M gven y: S S 5.34 X = X y := S X + jx j y.

10 Propertes: S I y = j y j, II S X + Y = S X+ S Y, III S fx = S f X + f S X. Metrzale SODE 171 It s easy to extend the acton of S to general vertcal tensor felds y requrng to preserve the tensor product. More precsely, we wll extend S to a specal class of tensor felds: Defnton 5.9. A d-tensor feld d from dstngushed on M s a tensor feld whose change of components, under a change of canoncal coordnates x, y x, ỹ on M, nvolves only factors of type x x and or x x. δ Example 5.2. δx and y are components of d-tensor felds of 1, 0- type. dx and δy are components of d-tensor felds of 0, 1-type, G are not components of a d-tensor feld snce a change of coordnates mples: 2 G = 2 x x j Gj ỹ x j yj ut t results that gven two semsprays 1 S and 2 S ther dfference X = 2 S 1 S s a vertcal vector feld. v j are not components of a d-tensor feld snce a change of coordnates mples: x j x k k = Ñ j x k k x + ỹj x. It follows that gven two nonlnear connectons 1 and 2 ther dfference F = 2 = 1 Fj = 2 j 1 s a d-tensor feld of 1, 1-type. j Defnton A metrc on M s a Remannan metrc g on the vertcal dstruton: g = g j x, y δy δy j. It results that g j are the components of a d-tensor feld of 0, 2-type wth the propertes: 1 g j = g y, y j, 2 symmetry g j = g j, 3 nondegeneraton det g j > 0. From the last property we derve the exstence of g 1 = g a x, y y a y whch s a d-tensor feld of 2, 0-type.

11 172 M. Crasmareanu Defnton The dynamcal dervatve of metrc g s S g : V M V M V M gven y: 5.35 S gx, Y = SgX, Y g S X, Y gx, S Y. The man noton of ths susecton s: Defnton The par S, s called metrc wth respect to g f: 5.36 S g = 0. The am of ths susecton s to detect all nonlnear connectons whch together wth S form a metrc par for a gven g. In order to answer at ths queston, a look at example 5.2 v gves necessary a study of two operators, called Oata n the followng, actng on the space of d-tensor felds of 1, 1-type: 5.37 O j kl = 1 δ 2 kδ jl gj g kl, The Oata operators are supplementary projectors: k 5.38 Oj a Ola = O a j O k la = 0, j Okl= 1 δ 2 kδ jl + gj g kl. OjO a k la = O k lj, a k O j Ola = O and tensoral equatons nvolvng these operators has solutons as follows: Proposton 5.9. The system of equatons: 5.39 a O j X ak = A jk, O a j Xak = A jk wth X as unknown has solutons f and only f: 5.40 O a A O a ak = 0, A ak = 0 j and then, the general soluton s: 5.41 Xjk = A jk + Oj a Y ak, X jk = A jk+ O wth Y an artrary d-tensor feld of 1, 1-type. j a j Y ak We are ready for one of the man results of ths paper whch s a natural generalzaton of Theorem 2.4 from [3]: Theorem 5.1. Set S and g. The famly S, g of all nonlnear connectons = j such that S, s metrc wth respect to g s gven y: c 5.42 j = 1 j ga g j c a ga S g aj + O a j X a wth X = X a an artrary d-tensor feld of 1, 1-type. It follows that S, g s a C M-affne module over the C M-module of d-tensor felds of 1, 1-type. k lj

12 Proof. We search j of the form: 5.43 j = c j +Fj Metrzale SODE 173 wth F j a d-tensor feld of 1, 1-type to e determned. The local expresson of equaton 5.36 s: 5.44 S g uv g um m v g mv m u = 0 and nsertng 5.43 n 5.44 gves: S g uv g um c m Multplyng the last relaton wth g ku yelds: 5.45 g ku S g uv c v g ku g mv k c m v g mv u = g um Fv m + g mv Fu m. c m u = F k v + g ku g mv F m u Let us verfy condton 5.40: Oav k g am S g m c a g am c g l lm = = g km S g mv c v g km g vl It follows: k Fj = 1 2 gm S g mj 1 j ga g j and returnng to 5.43 gves the concluson. c l k = 2 O av F a. m g km S g mv + g km c g vl lm + c v= 0. c c a +O aj X a In the spray case the equaton 5.42 admts a smplfcaton n wrtng: Proposton Fx a spray S and a metrc g. The famly S, g s: c 5.46 j = 1 j ga g j a ga y m δg aj δx m + Oa j X a. c A natural prolem s the varatons of S, g to varous changes of S and/or g. We treat here only the well-known case of conformal transformatons: Corollary 5.5. Let f C M, f > 0 everywhere on M. Then S, g = S, fg f and only f f s a frst ntegral of S.e. y f f = 2G x y Metrzalty of nonlnear connectons. Fx a nonlnear connecton = j and assocate to the semspray 3.12 whch we wll denote S. k Defnton The nonlnear connecton s metrc wth respect to g f the par S, s so. Theorem 5.2. The nonlnear connecton s metrc wth respect to g f and only f for all, j {1,..., n}: v 5.47 Ouj v u + g ua g v a + g um g mv y ay a = g m g mj x a ya.

13 174 M. Crasmareanu Proof. From 5.42 t results that s metrc f: v Ouj v u + g ua g v a + g um g mv y ay a = O v uj g um g mv x a ya and a straghtforward computaton of the rght-hand-sde of last equaton yelds the concluson. Example 5.3. Remannan metrcs: suppose g = g x. The last relaton ecomes: v 5.48 Ouj u v + g ua g v a = g m g mj x a ya whch s equvalent wth: 5.49 j + g a g j a = g m g mj x a ya. Let us consder that M s the tangent undle T and g s a Remannan metrc on. The Lev-Cvta connecton of g s a lnear connecton on M. A symmetrc lnear connecton wth coeffcents yelds a semspray S wth: Γ jk 5.50 G = 1 2 Γ jky j y k. The canonc nonlnear connecton of ths semspray has the coeffcents: 5.51 j = Γ jay a. Insertng 5.51 n 5.49 and neglectng y a gves: 5.52 Γ ja + g u g jv Γ v ua = g m g mj x a. But multplyng last equaton wth g s we get: 5.53 g s Γ ja + g j Γ sa = g sj x a whch s the usual Chrstoffel process. So, we verfed the condton 5.49 n the Remannan settng Metrzalty of semsprays. Defnton The semspray S s called metrc wth respect to g f the par S, c s metrc wth respect to g. Insertng c n the left-hand-sde of 5.42 we get: Theorem 5.3. The semspray S s metrc wth respect to g f and only f, for all, j {1,..., n}: v c u 5.54 Ouj v +g ua c g v a g um S g mv = 0. Corollary 5.6. The spray S s metrc wth respect to g f and only f, for all, j {1,..., n}: v c u 5.55 Ouj v +g ua c g v a g um y a δg mv δx a = 0.

14 Metrzale SODE 175 Example 5.4. Eucldean metrcs. Let us consder agan M as the tangent undle T and g s a constant metrc on.e. g j does not depend of x. The condton 5.54 s: v c u 5.56 O v +g ua c g v a = 0 whch means: 5.57 uj c j +g a g j c a= 0. If = R n and g s the usual Eucldean metrc then 5.57 reads: 5.58 c.e. the matrx j c j j + c = 0 elongs to o n=the Le algera of skew-symmetrc matrces of order n. So, we arrve at the well-known result that the orthogonal group On s the structural group of the Eucldean geometry on R n. Acknowledgement. I want to thank the referee for ponted out an mprovement of the text and Assoc. Prof. Petrcă Pop Edtoral Secretary of ths journal for excellent collaoraton. Also, I am deeply ndeted to my colleague Ioan Bucataru for several stmulatng suggestons and to referees for ponted out some mprovements over prevous versons. REFERECES [1] Brckell, F. and Clark, R.S., Integrale almost tangent structures, J. Dff. Geom , , MR #1163 [2] Bucataru, I., The Jaco felds for a spray on the tangent undle, ov-sad J. Math , o. 3, 69-78, MR k:53164 [3] Bucataru, I., Metrc nonlnear connectons, Dfferental Geom. Appl., , o. 3, , MR e:53141 [4] Clark, R. S. and Bruckhemer, M., Sur les structures presque tangents, C.R.A.S. Pars, , , MR #5983 [5] Clark, R. S. and Goel, D. S., On the geometry of an almost tangent manfold, Tensor, , , MR #4956 [6] Closs, M. P., On real almost hermtan structures suordnate to almost tangent structures, Canad. Math. Bull., , , MR #4743 [7] Closs, M. P., Remannan structures suordnate to certan almost tangent structures, Canad. Math. Bull , 71-77, MR #9463 [8] Crampn, M., Defnng Euler-Lagrange felds n terms of almost tangent structures Phys. Lett. A, , o. 9, , MR k:58072 [9] Crampn, M. and Thompson, G., Affne undles and ntegrale almost tangent structures, Math. Proc. Cam. Phl. Soc., , 61-71, MR g:53039 [10] Elopoulos, H. A., Structures presque tangents sur les varétés dfférentales, C. R. A. S. Pars, , [11] Elopoulos, H. A., Eucldean structures compatle wth almost tangent structures, Acad. Roy. Belg. Bull. Cl. Sc., , o. 5, , MR #3968 [12] Elopoulos, H. A., On the general theory of dfferentale manfolds wth almost tangent structure, Canad. Math. Bull., , , MR #1809, MR #1809

15 176 M. Crasmareanu [13] Grfone, J., Structure presque-tangente et connexons I, II, Ann. Inst. Fourer Grenole, , o. 13, , , MR #1409, MR #6112 [14] Krupkova, O., Varatonal metrcs on R T M and the geometry of nonconservatve mechancs, Math. Slovaca, , , MR a:53091 [15] Krupkova, O., Varatonal metrc structures, Pul. Math. Derecen, , o. 3-4, , MR k:53114 [16] Mestdag, T., Szlas, J. and Tóth, V., On the geometry of generalzed metrcs, Pul. Math. Derecen, , o. 3-4, , MR :53111 [17] Mron, R. and Anastase, M., The geometry of Lagrange spaces: theory and applcatons, Kluwer Academc Pulshers, FTPH o. 59, 1994, MR f:53120 [18] Szlas, J. and Muznay, Z., onlnear connectons and the prolem of metrzalty, Pul. Math. Derecen, , o. 1-2, , MR f:53040 [19] Vasman, I., Lagrange geometry on tangent manfolds. Int. J. Math. Math. Sc., , , math.dg/ MR k:53116 [20] Yano, K. and Daves, E. T., Dfferental geometry on almost tangent manfolds, Ann. Mat. Pura Appl. 4, , , MR #11781 UIVERSITY AL. I. CUZA DEPARTMET OF MATHEMATICS , IAŞI, ROMAIA MCRASM E-mal address: mcrasm@uac.ro

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