Bacso, Sandor

Berezovski, Volodymyr

Mikes, Josef

Miskolc Mathematical Notes

V. E. Berezovski

S. Bácsó

J. Mikes

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Diffeomorphism of affine connected spaces which preserved Riemannian and Ricci curvature tensors

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Miskolc Mathematical Notes HU e-ISSN 1787-2413Vol. 18 (2017), No. 1, pp. 117–124 DOI: 10.18514/MMN.2017.1978DIFFEOMORPHISM OF AFFINE CONNECTED SPACES WHICHPRESERVED RIEMANNIAN AND RICCI CURVATURE TENSORSV. E. BEREZOVSKI, S. BA´CSO´, AND J. MIKESˇReceived 27 April, 2016Abstract. In this paper we study the preserving of Riemannian and Ricci tensors with respect toa diffeomorphism of spaces with affine connection. We consider geodesic and almost geodesicmappings of the first type. The basic equations of these maps form a closed system of Cauchytype in covariant derivatives. We determine the quantity of essential (substantial) parameters onwhich the general solution of this problem depends.2010 Mathematics Subject Classification: 53B20; 53B21; 53B30; 53C26Keywords: preserving, Riemannian tensor, Ricci tensor, manifold with affine connection1. INTRODUCTIONThe theory of geodesic mappings was developed in the works of T. Thomas, H.Weyl, L.P. Eisenhart, P.A. Shirokov, K. Yano, A.Z. Petrov, A.S. Solodovnikov, N.S.Sinyukov, A.V. Aminova, J. Mikesˇ, I.G. Shandra, S.E. Stepanov and others, see [10–12, 14–16, 18–20, 22, 24].The problems raised in the study of geodesic maps were developed by V.F. Kagan,V. Vranceanu, P.K. Rashevsky, L.Ya. Shapiro, V.D. Vedenyapin, A.Z. Petrov andothers. In particular, the concept of quasi-geodesic mapping was introduced by A.Z.Petrov. Close to it there are holomorphically projective mappings of Ka¨hler spaces,considered originally by T. Otsuki and Y. Tashiro. See [15, 17, 19, 20, 23].It natural generalization of these classes there are almost geodesic mappings intro-duced by N.S. Sinyukov [20]. Recently investigated almost geodesic mappings V.S.Shadny, V.S. Sobchuk, N.V. Yablonska, J. Mikesˇ, V.E. Berezovski, M.S. Stankovic,L.S. Velimirovic, M.L. Zlatanovic [1–9, 19, 21].2. BASIC CONCEPTS OF THE THEORY OF DIFFEOMORPHISM OF AFFINECONNECTED SPACES AND THEIR TYPESWe consider an n-dimensional torsion-free affine connected space in the coordin-ate system .x/, and assume that n 2, further on all functions are sufficiently smooth.The paper was supported by the project IGA PrF 2014016 Palacky´ University Olomouc.c 2017 Miskolc University Press118 V. E. BEREZOVSKI, S. BA´CSO´, AND J. MIKESˇWe can find the basic concepts of the theory of geodesic and of almost geodesic map-pings in [15, 19, 20].Suppose, that an affine connected space An admits a diffeomorphism f onto anaffine connected space NAn in a common coordinate system x D .x1; : : : ;xn/.Let usP hij .x/D N  hij .x/   hij .x/ (2.1)given, where N  hij .x/ and   hij .x/ are components of objects of connections of affineconnected spaces An and NAn respectively in common coordinate system .x/, P hij is athe deformation tensor.Definition 1. A curve, defined in an n-dimensional affine connected space An, iscalled geodesic, if its tangent vector is parallel along it.Definition 2. A curve, defined in an affine connected space An .n > 2/, is calledalmost geodesic, if exists along it a two-dimensional parallel plane containing itstangent vector.Definition 3. Diffeomorphism f between two manifolds An and NAn with affineconnections is called geodesic mapping if any geodesic curve of An is mapped ontoa geodesic curve of NAn and vice versa.Diffeomorphism f is geodesic mapping if and only if the deformation tensor P hijin a common coordinate system has the following formsP hij .x/D  i .x/ıhj C j .x/ıhi ; (2.2)where ıhi is Kronecker symbol and  i is a covariant vector field.If  i  0, then f is an affine mapping.Definition 4. Diffeomorphism f : An ! NAn .n > 2/ is called almost geodesicmapping, if any geodesic curve ofAn is mapped onto an almost geodesic curve of NAn.Diffeomorphism f : An! NAn is almost geodesic if and only if the deformationtensor P hij in a common coordinate system has the following formsAh˛ˇ˛ˇ D a P h˛ˇ˛ˇ Cb h; (2.3)whereAhijk D P hij;kCP ˛ijP h˛k; (2.4)N  hij .x/ and   hij .x/ are affine connections of spaces Anand NAn, h is an arbitraryvector, a and b are some functions dependent on xh and h.Here and after the symbol “ ; ” means covariant derivation in An.N.S. Sinyukov [20] defined three types of almost geodesic mappings 1;2;3from the basic equations. It is well known [3, 6, 8], if n > 5 there exist only thesethree types.DIFFEOMORPHISM OF AFFINE CONNECTED SPACES 119Almost geodesic mapping of type 1 is characterized by the conditionsAh.ijk/ D a.ij ıhk/Cb.iP hjk/; (2.5)where aij is some symmetric tensor, bi is some covector, the bracket .ijk/ denotesthe symmetrization of the indices i;j;k (without division).If in (2.5) the covector bi is equal to zero, then the mapping is called cannonical. Itis well known, that arbitrary almost geodesic mapping of type 1 can be representedin the form of a composition of a canonical almost geodesic mapping of type 1 anda geodesic mapping [20].3. CONDITION OF REPRESENTATION OF RIEMANNIAN AND RICCI TENSORSWITH RESPECT TO THE DIFFEOMORPHISM OF THE AFFINE CONNECTED SPACEFrom equations (2.1) we obtain a relation between Riemannian tensors RhijkandNRhijkof spaces An and NAn respectively:NRhijk DRhijkCP hiŒk;j CP h˛ŒjP ˛ki ; (3.1)where Œjk denotes alternation of the indices j;k (without division).Then the relation (3.1) can be written asNRhijk DRhijk  AhiŒjk: (3.2)Therefore, we haveTheorem 1. The Riemannian tensor is preserved under diffeomorphism f : An!NAn if and only ifAhijk D Ahikj : (3.3)Contracting by indices h and k we obtainNRijk DRij  AhiŒjk: (3.4)where Rij and NRij are Ricci tensors of spaces An and NAn respectively.So we getTheorem 2. The Ricci tensor is preserved under diffeomorphism f : An! NAn ifand only if tensor Ahijksatisfies the following conditionsA˛ij˛ D A˛i j˛ : (3.5)120 V. E. BEREZOVSKI, S. BA´CSO´, AND J. MIKESˇ4. PRESERVATION OF THE RIEMANN AND RICCI TENSORS WITH RESPECT TOTHE GEODESIC MAPPINGS OF THE AFFINE CONNECTED SPACEConsider the geodesic mappings which are characterized by the equations (2.2).Form such mappings tensors of the formAhijk D ıhi  j;kC ıhj  i;kC2ıhk i j C ıhi  j kC ıhj  i k : (4.1)On the basis of Theorem 1 Riemannian tensor Rhijkis invariant with respect togeodesic mapping, if condition (3.3) satisfies.Substituting (4.1) into (3.3) we haveıhj . i;k   i k/C ıhi . j;k   j k/  ıhi . k;j   k j /  ıhk. i;j   i j /D 0:(4.2)After contraction (4.2) by indices h and i we obtain j;k   k;j D 0: (4.3)Transvecting (4.2) by indices h and j , we getn. i;k   i k/  . i;k   k i /D 0: (4.4)From (4.3) and (4.4) follows i;k D  i k (4.5)ThereforeTheorem 3. An affine connected space An admits geodesic mapping into an an-other affine connected space NAn preserving Riemmanian tensor if and only if theCauchy type system (4.5) has a solution with respect to functions  i .x/.On the basis of Theorem 2, the Ricci tensor Rij of an affine connected space Anis invariant with respect to geodesic mappings if satisfies condition (3.5).Substituting (4.1) into (3.5) we get i;j   j i D n. i;j   i j /: (4.6)From (4.5) and (4.6) we obtain (4.4).The above leads toTheorem 4. An affine connected space An admits geodesic mapping into an an-other affine connected space NAn preserving the Ricci tensor if and only if Cauchytype system (4.4) has a solution with respect to functions  i .x/.Integrability conditions of equations (4.4) are the following aRhijk D 0: (4.7)The general solution of Cauchy type system (4.4) depends on no more than n sub-stantial parameters.From (2.3) and (2.4) followsDIFFEOMORPHISM OF AFFINE CONNECTED SPACES 121Theorem 5. If under a geodesic mapping of an affine connected space An intoan affine connected space NAn Ricci tensor is an invariant geometric object, thenRiemannian tensor also is an invariant geometric object.This result can be achieved, if we consider the Weyl tensor.W hijk DRhijkC1nC1ıhi RŒjk 1n2 1Œ.nRij CRj i /ıhk   .nRikCRki /ıhj ; (4.8)which is an invariant object under geodesic mappings.SpaceAn which admits geodesic mapping preserving Riemannian and Ricci tensors,is characterized by the following interesting property.Theorem 6. In an affine connected space which admits geodesic mappings pre-serving Riemannian and Ricci tensors, there exists a covariant constant vector field.Indeed, from (4.3) we have that covector  i is locally gradient, that is, there existsfunction 	.x/ such that  i .x/D @i	.x/ where @i D @=@xi .We introduce function ˚.x/ D e 	.x/ and we can see on the basis of condition(4.5) satisfies ˚i;j D 0. Thus covector field 'i D @i˚.x/ is a covariant constant inspace An that is'i;j D 0: (4.9)On the other hand, if inAn there exists a covariant constant vector field 'i , then usingof the substitution 	.x/D  ln.˚.x//, @i˚.x/D '.x/ we can construct vector field i .x/, which satisfies (4.5) Thus Theorem 6 is proved.In addition, we note that equation (4.9) forms linear system of Cauchy type in co-variant derivatives with respect to unknown functions 'i .x/. Integrability conditionsof (4.9) has the form '˛R˛ijk D 0. There are linear algebraic equations with respectto 'i .x/. Their differential prolongations are also linear.The global existence of these fields was discussed in [13].5. PRESERVATION OF RIEMANN AND RICCI TENSORS WITH RESPECT TO THESPECIAL CASE OF CANONICAL ALMOST GEODESIC MAPPINGS OF THE FIRSTTYPE OF AFFINE CONNECTED SPACEConsider the special case of canonical almost geodesic mappings, which is char-acterized by the following equationP hij;kCP ˛ijP h˛k D ıh.iajk/; (5.1)where aij is some symmetric tensor.Indeed, mappings which are characterized by equation (5.1), are canonical almostgeodesic mappings of the first type, namely after symmetrization of equations (5.1)we obtain (2.5).122 V. E. BEREZOVSKI, S. BA´CSO´, AND J. MIKESˇFor mappings, which are characterized by equations (5.1), tensor A has the fol-lowing structureAhijk D ıh.iajk/; (5.2)Then, obviously, conditions (3.3) and (3.5) are fulfilled for the tensor A. Therefore,there is aTheorem 7. Riemannian and Ricci tensors are invariant geometrical objects un-der canonical almost geodesic mappings of the first type, which are characterized by(5.1).Because in a projective-euclidean space Riemannian tensor vanishes, and in aRicci-flat space Ricci tensor vanishes, then by Theorem (3.2) we haveTheorem 8. If projective euclidean or Ricci-flat space admits canonical almostgeodesic mapping of the first type, which is characterized by equations (5.1), intospace NAn, then NAnis projective-euclidean or Ricci-flat space respectively.Considering (5.1) as a Cauchy-type system with respect to deformation tensor P hijwe can determine their condition of integrability.For this we covariantly differentiate (5.1) on xm, then we alternate it by indices kand m. Using of the Ricci identity we haveıhi aj Œk;mC ıhj aiŒk;mC ıhŒkjij jm D P ˛ijRh˛kmCP h˛.jR˛i/kmCaj ŒmP hki Caj ŒmP hkj : (5.3)Contracting condition of integrability (5.3) by indices h and m, we getajk;i Caik;j   .nC1/aij;k D P ˛ijR˛kmCP ˇj˛R˛jkˇ Caj˛P ˛ki  ajkP ˛˛i Cai˛P ˛jk  aikP ˛j˛: (5.4)We alternate equations (5.4) by indices j and k, then we obtainaij;k D aik;j C 1.nC2/.P˛ijR˛kCP ˛ikR j˛  P ˇj˛R˛ikˇ CP ˇ˛kR˛ijˇ P ˇ˛iR˛jkˇ CP ˇ˛iR˛kjˇ  aj˛P ˛ki Cak˛P ˛ij CaikP ˛j˛ aijP ˛k˛/: (5.5)Observing (5.5), we can write equations in the following formaik;j D 1.n 1/.nC2/ .n.P˛ikR j˛  P ˇ˛.kR˛i/jˇ /CR˛.kP ˛i/jR˛ijˇ  P ˇj˛R˛.ik/ˇ P ˇ˛.iR˛jj jk/ˇ C .nC1/.aj.iP ˛k/˛ a˛.iP ˛k/j /C2.aikP ˛j˛ aj˛P ˛ik//:(5.6)DIFFEOMORPHISM OF AFFINE CONNECTED SPACES 123Obviously, equations (5.1) and (5.6) in a given space represent a Cauchy type sys-tem with respect to functions P hij .x/ and aij .x/ which, of course, are satisfied evenalgebraic conditions.P hij .x/D P hji .x/; aij .x/D aj i .x/: (5.7)So we provedTheorem 9. An affine connected spaceAnadmits almost geodesic mapping, definedby equation (5.1) onto affine connected space NAnif and only if there exist solution ofmixed equation system of Cauchy type (5.1), (5.6) and (5.7) with respect to functionP hij .x/ and aij .x/.The general solution of Cauchy type mixed system depends on (no more than)12n.nC1/2 real parameters.REFERENCES[1] V. Berezovski, N. Guseva, and J. Mikesˇ, “On special first-type almost geodesic mappings of affineconnection spaces preserving a certain tensor,” Mat. Zametki, vol. 98, no. 3, pp. 463–466, 2015,doi: 10.4213/mzm10817.[2] V. Berezovski and J. Mikesˇ, “On canonical almost geodesic mappings of the first type of affinelyconnected spaces,” Russ. Math., vol. 58, no. 2, pp. 1–5, 2014, doi: 10.3103/S1066369X14020017.[3] V. Berezovski and J. Mikesˇ, “Almost geodesic mappings of spaces with affine connection,” J.Math. Sci., New York, vol. 207, no. 3, pp. 389–409, 2015.[4] V. Berezovski, J. Mikesˇ, and A. 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Chuda´, “On geodesic mappings with certain initial conditions,” Acta Math. Acad.Paedagog. Nyha´zi. (N.S.), vol. 26, no. 2, pp. 337–341, 2010.[15] J. Mikesˇ, A. Vanzˇurova´, and I. Hinterleitner, Geodesic mappings and some generalizations.Olomouc: Palacky´ University, Faculty of Science, 2009.[16] J. Mikesˇ, “Geodesic mappings of affine-connected and Riemannian spaces,” J. Math. Sci., NewYork, vol. 78, no. 3, pp. 311–333, 1996.[17] J. Mikesˇ, “Holomorphically projective mappings and their generalizations,” J. Math. Sci., NewYork, vol. 89, no. 3, pp. 1334–1353, 1998.[18] J. Mikesˇ and V. Berezovski, “Geodesic mappings of affine-connected spaces onto Riemannianspaces,” in Differential geometry and its applications. Proceedings of a colloquium, held in Eger,Hungary, August 20-25, 1989, organized by the Ja´nos Bolyai Mathematical Society. Amsterdam:North-Holland Publishing Company; Budapest: Ja´nos Bolyai Mathematical Society, 1992, pp.491–494.[19] Mikesˇ J. at al, Differential geometry of special mappings. Olomouc: Palacky´ University, Facultyof Science, 2015.[20] N. Sinyukov, “Geodesic mappings of Riemannian spaces,” 1979.[21] M. S. Stankovic´, M. L. Zlatanovic´, and N. O. Vesic´, “Basic equations of G-almost geodesic map-pings of the second type, which have the property of reciprocity,” Czech. Math. J., vol. 65, no. 3,pp. 787–799, 2015.[22] E. S. Stepanova, J. Mikesˇ, and I. I. Tsyganok, “A geodesic mapping and its field ofsymmetric linear endomorphisms,” Diff. Geom. Appl., vol. 35, pp. 44–49, 2014, doi:10.1016/j.difgeo.2014.03.006.[23] K. Yano, “Differential geometry on complex and almost complex spaces,” Oxford-London-NewYork-Paris-Frankfurt: Pergamon Press. XII, 323 p. (1965)., 1965.[24] M. L. Zlatanovic´, L. S. Velimirovic´, and M. S. Stankovic´, “Necessary and sufficient conditionsfor equitorsion geodesic mapping,” J. Math. Anal. Appl., vol. 435, no. 1, pp. 578–592, 2016, doi:10.1016/j.jmaa.2015.10.052.Authors’ addressesV. E. BerezovskiUman National University of Horticulture, Dept. of Mathematics, Instytutska 1, Uman, Ukraine,E-mail address: berez.volod@rambler.ruS. Ba´cso´University of Debrecen, 4010 Debrecen, Hungary,E-mail address: bacsos@unideb.huJ. MikesˇPalacky University, Dept. of Algebra and Geometry, 17. listopadu 12, 77146 Olomouc, Czechia,E-mail address: josef.mikes@upol.cz

DIFFEOMORPHISM OF AFFINE CONNECTED SPACES WHICH PRESERVED RIEMANNIAN AND RICCI CURVATURE TENSORS

DIFFEOMORPHISM OF AFFINE CONNECTED SPACES WHICH PRESERVED RIEMANNIAN AND RICCI CURVATURE TENSORS

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