η-einstein Complex Finsler-Randers Spaces

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1 Open Access Journal of Matematcal and Statstcal Analyss RESEARCH ARTICLE η-ensten Complex Fnsler-Randers Spaces Roopa MK * and Narasmamurty SK Department of PG Studes and Researc n Matematcs Kuvempu Unversty Karnataa Inda * Correspondng autor: Roopa MK Department of PG Studes and Researc n Matematcs Kuvempu Unversty Sanaragatta Svamogga Karnataa Inda Tel: E-mal: roopam100@gmalcom Ctaton: Roopa MK Narasmamurty SK (019) η-ensten Complex Fnsler-Randers Spaces J Mat Stat Anal 1: 301 Abstract In ts paper we determn te necessary and suffcent condtons tat a complex Randers metrcs sould be η-ensten and wt te condton of Ka ler we prove tat η-ensten complex Randers spaces of nonzero olomorpc curvature are purely Hermtan Furter we prove a Scur type lemma for η-ensten complex Randers space Also we prove te η- Ensten complex Randers space of non-zero constant olomorpc curvature are wealy Ka ler And some examples sows η-ensten purely Hermtan complex Fnsler-Randers metrc Keywords: Complex Fnsler Space; Complex Randers Metrc; η-ensten Metrc; Remann Curvature; Holomorpc Curvature; Ka ler Introducton Te real Randers metrc were frst ntroduced by G Randers n te context of general relatvty and tey were appled to te teory of te electron mcroscope by R Ingarden [1] As compared to te real case n complex Fnsler geometry tere are not so many nown classes of complex Fnsler metrcs Besdes te sgnfcant Kobayas and Carateodory metrcs te study of suc Fnsler geometry tey are trval classes of complex Fnsler metrcs: Te complex Fnsler metrcs wc comes from Hermtan metrcs on te base manfold and locally Mnows complex metrc In 009 te autors Ncoleta Aldea and George Munteanu were studed te complex Fnsler spaces wt Randers space and Ka ler Randers metrc see [] Te complex Fnsler metrcs of constant olomorpc curvature s an nterestng problem n complex Fnsler geometry Te autor M Abate and G Patrzo were gave te caracterzaton of te constant olomorpc curvature troug complex geodescs and obtaned te complex Fnsler metrcs of olomorpc curvature K F = 4 and wc satsfes some regularty condtons of Kobayas metrc Also tey proved tat f te complex Fnsler spaces satsfy te noton of Ka ler a symmetry condton on te curvature and wt postve constant olomorpc curvature ten t s a purely Hermtan In Ncoleta Aldea as determned te condton n wc complex Fnsler metrcs as constant olomorpc curvature on te bass of generalzed Ensten and ntroduced a new class of complex Fnsler metrcs s called η-ensten [3] Regardng tese concepts n te present paper we sall determne te necessary and suffcent condtons tat a complex Randers metrcs sould be η-ensten and wt te condton of Ka ler we proved tat η-ensten complex Randers spaces of non-zero olomorpc curvature are purely Hermtan Furter we prove a Scur type lemma for η-ensten complex Randers space Prelmnares Let M be an n-dmensonal complex manfold and z = (z ) =1n be complex co-ordnates n a local cart Te complexfed of te real tangent bundle T c M splts n to te sum of olomorpc tangent bundle T' M s tself a complex manfold and local co-ordnates n a local cart wll be denoted by u = (z η ) =1 n Tese are canged n to ( z η ) by te rules = 1 n z = η ( z) and z l η = η l z A complex Fnsler space s a par (M F) were F: T' M R + s contnuous functon satsfyng te condtons: () L = F s smoot on T M = T M / {0} () F(z η) 0 te equalty olds ff η = 0 () F( z λη) = λ ( z η) for λ (v) te Hermtan matrx ( g ( z η )) s postve defnte were g L = s te fundamental metrc tensor η η ScolArena wwwscolarenacom

2 J Mat Stat Anal Equvalently t means tat te ndcatrx s strongly pseudo-convex L L g η = η = L η = 0 and L = g ηη Consequently from () we ave We say tat a functon f on T' M s (p q)- p q omogeneous wt respect to η ff f( z λη) = λ λ f( z η) λ By Euler s teorem ts omogenety condton s equvalent to f and f for nstance L = F η = pf η = qf s a (11) omogeneous functon and te complexfed tangent bundle of T' M s decomposed n T ( T M) = T ( T M) T ( T M) m δ g m g Were m m L = g C = g δz Te covarant dervatve of Wt C Consderng te restrcton of te proecton to T M = T M / {0} for pullng te olomorpc tangent bundle T' M bac ten * we obtan te olomorpc tangent bundle π : π ( T M) ( T M) s called te pullbac tangent bundle over te slt ( T M) we * denote by * * are local frame and ts dual tangent bundle co-ordnates respectvely a { nd dz dz } z z Let V( T M) = er π* T ( T M) be te vertcal bundle locally spanned by complex non-lnear connecton (cnc) determnes a supplementary complex sub bundle to VT ( M) n T ( T M) e T ( T M) = H( T M) V( T M) Te adapted frames of te (cnc) δ = N were N ( z η ) are te coeffcent of te (cnc) Furter we use te abbrevatons and ter conugates as δz n δ δ δ = = δ = and = δz δz Let g be te fundamental metrc tensor of complex Fnsler space (M L = F ) [-4] Te somorpsm between π (T' M) and T' M nduces an somorpsm of π (T c M) and (T c M) Tus g * * * defnes an Hermtan metrc structure G( z η) = g dz dz on π ( Tc M ) Wt respect to te natural complex structure Moreover H(T' M) and π (T' M) are somorpc Terefore te structure on π (T c M) can be pulled bac to H( TM ) H( TM ) By ts somorpsm te natural co-bass dz * s dentfed wt dz In vew of tese constructons te connecton of G ts s of te form [5] w ( z η) = L ( z η) dz + C ( z η) δη * X = X ( z η) assocated to te Cern lnear connecton (clc) s z ( δη δη ) X X dz + X + X dz + X * z = X = X + XL X = X + XC X = δ X X = X Te Cern-Fnsler (clc) on π (T' M) determnes te Cern (cnc) on (T' M) wt te coeffcents coeffcents of torson and curvature are T = L L N = g g z η m m and ts local l R = δ L δ NC Ξ = δ C =Ξ l P = L C S = C = S l l ScolArena wwwscolarenacom

3 J Mat Stat Anal 3 Te Remann tensor RWZXY ( ) = GRXYWZ ( ( ) ) as te propertes: l RW ( Z X Y) = WZ X Y R R = R g R R R R l = = = By settng R = R η = g N ι ι δ η te Rcc scalar and te Rcc tensor assocated to te Cern (clc) on π (T' M) are defned by Rc Rc = g R = R ηη Rc = ten t sows tat are 1-omogeneous wt respect to R R η η η e η = R l Accordng to te complex Fnsler space (M F) s strongly Ka ler T = 0 and wealy Ka ler ff g Tηη = 0 l note tat for a complex Fnsler metrc wc comes from a Hermtan metrc on M so called purely Hermtan metrc [1] Te olomorpc curvature of F n drecton η wt respect to te Cern (clc) s gven by R( ηηηη ) ηηr KF ( z η) = = G L ( z η) ( ηη) (1) were η s vewed as local secton of π * ( T M) e η = η z Furter we sall smply call t as olomorpc curvature It depends bot on te poston z M and te drecton η Moreover t s 0-omogeneous wt respect to η In Vew of [6] Ncoleta Aldea as ntroduced te followng concepts [36]: Defnton 1 Te complex Fnsler space (M F) s called generalzed Ensten f real valued functon K(z η) suc tat * R s proportonal to t e f tere exsts a R K( z η) t () L L were t = Lz ( η) g + ηη η = η = Also tey were proved tat te man propertes of g-ensten complex Fnsler space as: Teorem 1 Let (M F) be g-ensten complex Fnsler space Ten 1 () K( z η) = K ( z η) 4 F and t depends on z-alone () If (M F) s connected and wealy Ka ler of complex dmenson n ten t s a space wt constant olomorpc curvature () If te space s non-zero constant olomorpc curvature ten F s wealy Ka ler (v) If te space s Ka ler of non-zero constant olomorpc curvature ten F s purely Hermtan ff t s Ka ler of constant olomorpc curvature e g-ensten Defnton Te complex Fnsler space (M F) s called η-ensten f tere exsts two smoot functons K (z η): K( z η ) : T M R = 1 suc tat Table of symbols n Complex Fnsler geometry R K z Lg K z = 1( η) ( ) + ηηη (3) Te below table lsts te basc symbols n complex Fnsler geometry and ter defntons (wen sort enoug to lst) However t sould be ponted out tat te quanttes and notaton n complex Fnsler geometry s far from standardzed Name Metrc tensor of Complex Fnsler space g Notaton L = η η Holomorpc tangent bundle (T' M) ScolArena wwwscolarenacom

4 4 J Mat Stat Anal proecton of Holomorpc tangent bundle ( T M) Hermtan metrc structure Gzη ( ) Pull bac tangent bundle π (T' M) Cartan tensor Cern-Fnsler connecton co-effcents C C Curvature coeffcents R R Rcc tensor Rc Te noton of complex Randers space Holomorpc curvature K ( z η) Coeffcents of Cern-nonlnear connecton N ( z η) Let F = α + β be complex Randers metrc were α( z η) = a ηη be a purely Hermtan postve metrc and β( z η) = β( z η) β( z η) wt β( z η) b ( z) η M F = α + β s called a complex Randers space = be a dfferentable 1-form and te par ( ) Te functon L= F = ( α + β ) depends on z and η by means of real valued functon α = α( z η) a nd β = β( z η) Moreover α and β are omogeneous wt respect to η e α( z λη) = λ α( z η) β ( z λη) = λβ ( z η) for any λ tus L( z λη) = λλl( z η) for any λ and so te omogenety property mples te followng quanttes; α 1 β 1 = α η = β L L F L L L α = = α β α + β = β αl + βl = Lαα L + βl = L αα αβ αβ β β β F α αα α β αβ β β L + L + L = L L L L Lα = L = Lαα = β Were α β α etc To determnes te fundamental metrc tensor g of te complex Randers space ( α + β ) usng te nvarants as; g = α 1 β β = l = b α β b = a b b = a bb γ = L+ α ( b 1 ) (31) L α β F Fβ η = = L L l b α + = = + β α β were l = a η and ( ) a s Hermtan nverse of ( a ) By usng te formula of (g ) n [9] and te nvarants (31) we obtan (3) Were ScolArena wwwscolarenacom

5 J Mat Stat Anal 5 Te nverse of fundamental metrc tensor (g ) s gven by Te complex Cartan tensors from we ave te Cartan tensor of complex Randers space as Were and wt n (3) Ten te vertcal coeffcents of Cern-Fnsler connectons are Solvng above we get [7] Were η-ensten Complex Randers-metrcs In ts secton we determne te necessary and suffcent condtons tat a complex Randers metrcs must be η-ensten Proposton 41 Let F = α + β be a (η-e) Complex Randers space of complex dmenson n Ten tey are followng ScolArena wwwscolarenacom

6 6 J Mat Stat Anal (v) te functons K (z η) = 1 are 0-omogeneous wt respect to η f n Proof In vew of (1) and usng (3) and contractng t wt ηη we obtan Ts proves () () Snce te functon R are 1-omogeneous wt respect to η wc mples tat t yelds () On te oter and usng () K F (z η) s 0-omogeneous wt respect to η e K F (z λη) = K F (z η) for any λ we ave We ave (41) Dfferentate above wt respect to λ and lettng λ = 1 we get Hence we ave ScolArena wwwscolarenacom

7 J Mat Stat Anal 7 Ts proves () By conugaton n (3) we ave wc gves (v) From condton () we ave (4) Equaton (4) can be wrtten as were ScolArena wwwscolarenacom

8 8 J Mat Stat Anal But g n 1 l = and n terefore Ln K z η η ( 1) ( ) = 0 and from ere we get l Above equaton K (z η) s 0-omogeneous wt respect to η and from () K 1 (z η) s also 0-omogeneous wt respect to η Ts fnses (v) Teorem 4 Let F = α + β be complex Randers space of complex dmenson n Ten te followng are equvalent () (M F = α + β ) s η-ensten () Tere exsts two smoot functons K 1 (z η) and K (z η) are 0-omogeneous wt respect to η and satsfes te followng; were () Tere exsts two smoot functons K 1 (z η) and K (z η) are 0-omogeneous wt respect to η and suc tat Proof Snce F s η-ensten Ten dfferentate R from proposton 41 wt respect to η and η l respectvely we ave ScolArena wwwscolarenacom

9 J Mat Stat Anal 9 (43) and (44) β α Were Ξ= η = l 1+ + bβ 1 + α β Tere exsts a functons K (z η) = 1 are 0-omogeneous wt respect to η we get (45) Now we prove te teorem: () () : Gven R as n (3) we can reconstruct ts R Usng te Banc dentty[4] On contractng ts wt ηη we obtan On te oter and Were R = R η Snce R = C η + R l l l It ndeed Rl = C η + R l l wc togeter wt (43) we get () Now from proposton 41 condton (v) te functons K (z η) = 1 are 0-omogeneous wt respect to η Conversely contractng te equaton from () by η we ave ScolArena wwwscolarenacom

10 10 J Mat Stat Anal (46) Snce K = ( z η) η = 0 = 1 and from (46) te last term becomes zero so (M F) s η-ensten () (): From we use te followng Fnsleran second Banc dentty to construct R as l On contract above wt η we obtan But so tat It mples tat (47) Usng te notatons form [5] and equaton (44) n to te above equaton we get value of proposton 41 te functons K (z η) = 1 are 0-omogeneous wt respect to η To prove te condton () frst we compute ( R R ) ηη l l l R l Furter condton (v) from (48) ScolArena wwwscolarenacom

11 J Mat Stat Anal 11 r l Snce C C ηη = C = 0 and 0 r l T = So we ave () [5] Proposton 4 Let F = α+ β be a η-ensten complex Randers space of dmenson n Ten () K F does not depends on η () () (v) Proof As we Know ten On contractng (47) by η l η and by teorem 4 condton () we deduce tat (49) On te oter and l R ηη l and from equaton (47) we ave On bot sde conugaton above we get (410) From equaton (49) and (410) t yelds ScolArena wwwscolarenacom

12 1 J Mat Stat Anal (411) To prove () we contract (411) by η we ave Wc mples tat ( K1+ K) = 0 e ( K1+ K) = 0 Apply conugaton above we get ( K1+ K) = 0 Clearly (K 1 + K ) does not depends on η Now usng () and equaton (411) ten t proves () To prove (): From Jacob dentty [5]: We ave Usng (31) we obtan On contractng (41) by glrη t yelds ScolArena wwwscolarenacom

13 J Mat Stat Anal 13 Ts proves () And obvously usng () and () we get (v) Corollary 41 Let F = α+ β be a η-ensten complex Randers space of dmenson n Ten () () s real valued Proof Snce Rc = nk 1 + K L Dfferentate ts wt respect to η we ave Ten But by proposton 4 condton () we ave K = K(z) K 1 It becomes K = K K = K and K = K Usng tese we get () Remar 43 Any η-ensten complex Fnsler space te olomorpc curvature does not depends on η but t depends on z only K F (z) = K F (z η) = K(z) and a η-ensten complex Fnsler space s g-ensten f K 1 = K Regardng tese we ave te followng: Corollary 4 If F = α+ β s a Ka ler η-ensten complex Randers space of dmenson n ten t s g-ensten Proof Snce F s Ka ler η-ensten complex Randers space By proposton 4 condton () we ave T η and so T = 0 we obtan wc mples tat ScolArena wwwscolarenacom

14 14 J Mat Stat Anal Terefore K 1 = K Form corollary 4 and teorem 31 we obtaned te followng lemma: Lemma 41 If F = α + β s a Ka ler η-ensten complex Randers space of dmenson n wt K(z) 0 ten F s purely Hermtan Proof Snce F s Ka ler η-ensten complex Randers space and note tat a complex Fnsler metrc s purely Hermtan f and only f C K = 0 and from teorem 41 we see tat te equaton (48) C = 0 and so F s purely Hermtan η-ensten complex Randers space wt constant Holomorpc curvature In ts secton we determne te condtons under wc -Ensten complex Randers space wt constant olomorpc curvature e K(z) = K 1 (z η) + K (z η) s constant By pursung ts frst we prove te Scur type lemma for η-ensten complex Randers space: Lemma 5 (Scur type lemma:) Let F = α+ β be a η-ensten connected complex Randers space and s wealy Ka ler Ten F as constant olomorpc curvature Proof From proposton 41dfferentate R wt respect to η l we ave 51 Usng te Banc dentty as Contracton above wt l grηηη we ave From ts and (51) gves 5 Snce F s wealy Ka ler ten from equaton (5) we get ScolArena wwwscolarenacom

15 J Mat Stat Anal 15 so by conugaton we ave 53 Because of K( z) = K( z) = 0 From equaton (53) we can easly see tat Multply above by g t yelds Now pluggng ts nto (53) wc follows tat It mples tat K(z) = constant Also from equaton (5) we deduce te followng; Proposton 53 If F = α + β s η-ensten complex Randers space wt K(z) s a non zero constant ten F s wealy Ka ler Proof Snce F s η-ensten wt K(z) s a non zero constant ten K(z) l = 0 and te equaton (5) yelds wc mples tat F s wealy Ka ler Remar 54 If (M F) s η-ensten complex Fnsler space wt K(z) = 0 ten t s a flat complex Fnsler space e K F = 0 and αβ C = CC = 0 From ts general case moreover usng teorem 41 we prove te followng: δ Teorem 55 Let F = α + β be a complex Randers space of dmenson n Ten te followng are equvalent () (M F) s η-ensten wt constant curvature K F = (K 1 + K ) = λ λ R () Tere exsts two smoot functons K 1 (z η) and K (z η) suc tat K1(z η) s 0-omogeneous wt respect to η K1(z ) + K(z ) = λ and ScolArena wwwscolarenacom

16 16 J Mat Stat Anal 54 Proof From teorem 41 we see tat F s η-ensten ten tere exsts te smoot functon K 1 and K are 0-omogeneous wt respect to η and satsfes te condton () Moreover K(z) = K 1 (z η) + K (z η) = λ and combnng te equaton from menton teorem condton () we obtan (54) and so () () Conversely contractng (54) by η and by tang K 1 (z η) + K (z η) = λ and K 1 (z η) s 0-omogeneous wt respect to t sows () Example: We gve an example wc sows te above secton teory η + z η < z η > < z η > L = + ( 1+ z ) ( 1+ z ) 55 n n defned on te ds 1/ n n be a complex Randers metrc were z = zz < z η >= zz = 1 = 1 r = z z < rr= n f < 0 on f = 0 and on te complex proectve space P n n ( ) f > 0 Notce tat α = a ( ) z ηη and tus determnes purely Hermtan metrcs wc ave specal propertes Tey are K aler wt constant olomorpc curvature K α = 4 [5] n n n In partcular for = 1 we obtan te Bergman metrc on te unt ds = 1 for = 0 te Eucldean metrc on and for = 1 te Fubn-Study metrc on P n n ( ) Tey are purely Hermtan It ndeed tey are well nown metrcs [8-1] Terefore te olomorpc curvature of (55) s α F KF = = 1 = 1 Lα (1 z ) Now let us consder a Fnsler metrc wc s conformal to (55) e Clearly g s purely Hermtan and by te equaton R (see secton-4) we ave Suppose and t s not Ka ler Furtermore we ave Wc sows tat ScolArena wwwscolarenacom

17 J Mat Stat Anal 17 Were So te metrc of (55) s η-ensten wt olomorpc curvature Ts example sow of η-ensten purely Hermtan complex-randers space but not Ka ler Concluson Te noton of complex Fnsler metrc s ntroduced by Carateodory and so called Carateodory metrc Te geometry of complex Fnsler manfold va tensor analyss was started by te autor G B Rzza and afterwords te connecton teory on complex Fnlser manfold as been developed by H Rund Y Icyo M Fuu et al Recently from vew pont of geometrc teory of several complex varables complex Fnsler metrc as became an nterestng subect In partcular Kobayas metrc and one more nterest n complex Fnsler geometry to study te olomorpc vector bundles As we now n general Ensten metrcs s sad to Rcc tensor s proportonalty of metrc tensor e Rc α g wc are a natural extenson of tose n Remannan geometry and tey ave good propertes n Remann geometry for some class of Fnsler metrcs [13-15] In Fnsler geometry frst advance of Ensten Fnsler metrc s Ensten Randers type by C Robels But n complex nsler geometry te autor Ncoleta Aldea as ntroduced η-ensten based on te olomorpc curvature Especally n ts study we fnd te necessary and suffcent condtons tat a complex Randers metrcs sould be Ensten and wt te condton of Ka ler we prove η-ensten complex Randers spaces of non-zero olomorpc curvature are purely Hermtan Furter we prove a Scur type lemma for η-ensten complex Randers space Also we prove te η-ensten complex Randers space of non-zero constant olomorpc curvature are wealy Ka ler References 1 Abate M Patrzo G (1994) Fnsler Metrcs A Global Approac In Wt Applcatons to Geometrc Functon Teory Lecture Notes n Matematcs 1591: 170 Aou T (003) Proectve Flatness of Complex Fnsler Metrcs Publ Mat Debrecen 63: Aldea N (005) Complex Fnsler spaces of constant olomorpc curvature Derental geometry and ts Applcatons Matfyzpress Prague Aldea N (005) On te curvature of te (ge) conformal complex Fnsler metrcs Bull Mat Soc Sc Mat Roumane 48: Aldea N (008) On olomorpc curvature of -Ensten complex Fnsler space Bull Mat Soc Sc Mat Roumane 51: Aldea N (00) On Cern complex lnear connecton Bull Mat Soc Sc Mat Roumane 45: Aldea N Munteanu G (007) -complex Fnsler metrcsproceedngs of te 4t Internatonal Colloquum Matematcs n Engneerng and Numercal Pyscs 1-6 BGS Proc 14 Geom Balan Press Bucarest 8 Aldea N Munteanu G (009) On complex Fnsler space wt Randers metrc J Korean Mat Soc 46: Aldea N Munteanu G (006) On te geometry of complex Randers spaces Proc of te 14-t Nat Sem on Fnsler Lagrange and Hamlton spaces Brasov Cen B Sen Y (007) Complex Randers Metrcs communcated to Conference Zeang Unv Cna 11 Ingarden RS (1957) On te geometrcally absolute optcal representaton n te electron mcroscope Trav Soc Sc Lett Wroclaw Ser B 3: 60 1 Munteanu G (004) Complex Spaces n Fnsler Lagrange and Hamlton Geometres Complex Spaces n Fnsler Lagrange and Hamlton Geometres Kluwer Acad Publ 13 Roopa MK Narasmamurty SK (016) Rcc Tensor of Fnsler Space wt specal -metrcs Acta Matematca Academae Paedagogcae Nyregyazenss 3: Sen Z (001) Dfferental Geometry of Spray and Fnsler spaces Dordrect Kluwer Academc Publsers 15 Spro A (001) Te Structure Equatons of a Complex Fnsler Manfold Asan J Mat 5: ScolArena wwwscolarenacom