Tikrit Journal of Pure Science 21 (1) 2016 ISSN:

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1 Tikrit Journl of Pure Siene () 06 ISSN: K- onstnt type Khler n Nerly Khler mnifols for onhrmoni urvture tensor Ali A. Shih, Rn H. jsim Deprtment of Mthemtis, College of Eution Pure Sienes, Tikrit University, Tikrit, Irq li.8@yhoo.om rn.hzim03@gmil.om Astrt The onstnt of permnene onhrmoni type khler n nerly khler mnifol onitions re otine when the Nerly Khler mnifol is mnifol onhrmoni onstnt type (K). n lso prove tht M nerly khler mnifol of pointwise onstnt holomorphi setionl onhrmoni (PHKm()) urvture) urvture tensor if the omponents of holomorphi setionl (HS- urvture) urvture tensor in the joint G-struture spe tht stisfies onition. Keywors. Conhrmoni onstnt type khler n nerly khler mnifols, mnifol pointwise onstnt holomorphi setionl onhrmoni. Definition() [5] Let (M, J, g) is NK- mnifol of imension n, K - tensor onhrmoni urvture. tht omponents tensor Riemnn- Christoffel on spe of the joint, G-struture will e Remine [3] look like: ) R 3) R 5) R 8) R ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ ˆˆ ˆˆ 0; ) R 0; 4) R 0; 6) R 0; 9) R ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ R R ˆ ˆ ˆˆ h h A A h h ; 7) R h ˆ ; 0) R h ˆ h ; h 0; 0. () n the omponents of Rii tensor S on spe of the joint G-struture look like: ) S 0; ) S ˆ 0; 3) S ˆ ˆ Sˆ A 3 h..() At lst slr urvture nerly Khler mnifols in the spe of the joint G-struture is lulte uner the formul χ = A + 6..(3) Proposition() [5] Let (M, J, g) - NK- mnifol. The urvture tensor onhrmoni ws introue will e Remine Ishii [] s tensor of type (4, 0) on n-imensionl Riemnnin mnifol, etermine y the formul K(,,,W) = R(,,,W) [g(, W)S(, ) (n ) g(,)s(,w) + g (, )S (,W) g (,W)S(,)] (4) Where R the Riemnn urvture tensor, S - Rii tensor. This tensore is invrint uner onhrmoni trnsformtions, i.e. with onforml trnsformtions of spe keeping hrmony of funtions. Let's onsier properties tensor onhrmoni urvture: [6] ) K,,, W,,, W g, W S, g, S, W n g, S, W g, W S, R,,, W n g, W S, g, S, W n g, S, W g, W S, K,,, W. n Properties re similrly prove: ) K(,,,W) = K(,, W, ). h ; ) K(,,,W) = K(,W,,). 3) K(,,,W) (,,,W) (,,,W) =0. Thus onhrmoni urvture tensor stisfies ll the properties of lgeri urvture tensor: ) K(,,,W) = K(,,,W); ) K(,,,W) = K(,, W, ); 3) K(,,,W) (,,, W) (,,, W) ; 4)K(,,,W) = K(, W,, );,,, W (M),where (M)is moule of vetor fiels on mnifol M (5) Lets Clulte Components of the Components tensor urvture on spe of the jonit G-struture for Nerly Khler mnifol, In terms of the ovrint omponents of the form[5] We shll write own s : K ijkl = R ijkl - ( gik δ (n ) jl + gjl δ jk - gjl δ jk - gjk δ il ).(6) we will extrt ompouns the tensor onhrmoni whih my extrte from the soure in nother wy, s in the following theory. Theorem(3) : The omponents of The onhrmoni onstnt onirulr tenser of Nerly Khler -mnifol in the jonit G-struture spe re given y the following forms: - K â =A - h h - δ + δ δ ).(7) - K ĉ = h h - δ δ (n ) (n ) δ + δ )+ + δ δ ) (8) n the other re onjugte to the ove omponents or equl to zero proof: K ijkl = R ijkl - (n ) (n ) ( gik δ jl + gjl δ jk - gjl δ jk - gjk δ il ) - Put i = j = k = n l = K = R - ( g δ + g δ - g δ - g δ ) - (n ) (0) 07

2 Tikrit Journl of Pure Siene () 06 ISSN: Put I = â j = k = n l = K â = R â - ( g â (n ) δ + g δ â - g â δ - g δ â ) - (n ) 3- put i =, j =, k = n l = K = R δ - g δ - - (n ) ) (0) (n ) ( g δ + g δ - g 4- Put i =, j =, k = n l = K = R - g δ - g δ - ) (0) (n ) (n ) ( g δ + g δ - 5- Put i =, j =, k = n l = K = R - (n ) ( g δ + g δ - g δ - g δ ) - (0) (n ) 6- Put i = j = k = n l = K = R â â - ( g (n ) g δ - g δ ) - δ â + g δ - (n ) 7- Put i = j = k = n l = K ĉ = R ĉ - ( g δ (n ) + g δ - g δ -g δ ) K ĉ = h h - δ + (n ) δ δ )+ δ + δ δ ) 8- Put i = j = k = n l = K â = R â - ( g (n ) â â - g â δ -g δ â ) δ δ ) K â =A - h h - δ + g δ (n ) δ + i.e for Nerly Khler, only two onhrmoni invrint on t equl to zero ; K = K () = -K with omponent {K, K } n K 4 = K (4) with omponent { K, K }, other omponents onhrmoni urvture Tensor equl to zero. Definition (4) : [4] Suppose tht λ(,,,w ) = K(,,, W) K(,, J, JW) Consier the following tensor λ(,) = λ(,,,) We sy tht n lmost hermition mnifol M is of onstnt type t p M Provie tht for ll T P (M),where T P (M) is tngent spe of M t the point p λ(,) = λ(,).( 9) Remrk(5): - If (9) hols for ll p M then the mnifol M hs pointwise onstnt type. - If (9)is onstnt funtion,then (M,J,g) hs glol onstnt type. Definition (6) : An lmost hermition mnifol M n onhrmoni type (K- onstnt type ) If,,,W (M n ) tht λ(,,,w ) = K(,,, W) K(,, J, JW) Consier the following tensor λ(,) = λ(,,,) We sy tht n lmost hermition mnifol M is of onstnt type t p M Provie tht for ll T P (M),where T P (M) is tngent spe of M t the point p λ(,) = λ(,) Theorem (7): If M is Nerly Khler mnifol of The onhrmoni tenser then M mnifol onhrmoni onstnt onirulr if n only if λ(,) = λ(,) = -8 h h - (n ) δ + δ δ )+ δ + δ δ ) Proof : suppose tht M is Nerly Khler mnifol of onhrmoni tenser At first we fin the following result y using efinition( 6) it follows :- - λ(, ) = K(,,, ) K(,, J, J) Let M n Nerly Khler mnifol To ompute the K(,,, )n K(,, J, J) on the spe of the jont G-struture (i) K(,,, ) = K ijkl i j k l = K +K ĉ +K ĉ â â â â ĉ â + K â ĉ â â ĉ + K ĉ â ĉ â ĉ y using the properties of onhrmoni tenser eqution (5) We get : K (,,, ) = K â â â ĉ â + K ĉ + K ĉ.(0) (II) K (,, J, J ) In the joint G-struture spe the omponents 08

3 Tikrit Journl of Pure Siene () 06 ISSN: K (,, J, J ) = K ijkl i j (J) k l = K (J) (J) ĉ (J) (J) ĉ (J) (J) ĉ â (J) ĉ â â (J) (J) â (J) â (J) â (J) â (J) ĉ â (J) ĉ y using the properties of onhrmoni tenser eqution (5) We get : K (,, J, J )= K â () () + K â (J) ĉ K â (J) K (J) (J) ĉ K (J) ĉ () Mking use of the eqution (0) n () we get : K(,,, ) K(,, J, J)= K â â â + K ĉ +K () ĉ () K â (J) â (J) +K -K â (J) ĉ - K (J) - K (J) ĉ â ( J) ĉ = - 4 K This is the K 4 of theory (3) eqution (8) n the ompenstion we get : = - 4[ h h δ + δ δ ) = - 8 h h - (n ) δ (n ) δ δ + δ δ ).() - λ(, ) = K(,,, ) K(,, J, J) + δ δ )+ + δ δ )+ To ompute the K(,,, )n K(,, J, J) on the spe of the jont G-struture ( I ) K(,,, ) = K ijkl i j k l = K +K ĉ + K ĉ â â â â ĉ â + K â ĉ K â ĉ â ĉ + ĉ â ĉ y using the properties of onhrmoni tenser eqution (5) We get : K (,,, ) = K â ĉ â ĉ + K â â + K ĉ..(3) (II) K (,, J, J ) In the joint G-struture spe the omponents K (,, J, ) = K ijkl i j (J) k l = K (J) (J) ĉ (J) (J) ĉ â (J) (J) ĉ â (J) ĉ â (J) (J) â (J) â (J) â (J) â (J) ĉ â (J) ĉ y using the properties of onhrmoni tenser eqution (5) We get : K (,, J, J )= K â â (J) ĉ â K (J) K ( J) (J) ĉ K (J) ĉ.(4) Mking use of the eqution (3) n (4) we get : + K(,,, ) K(,, J, J)= K â â K â ĉ â ĉ â â ĉ ĉ () ĉ () K â â - K - K ĉ (J) ĉ ĉ â (J) ĉ - K â â (J) â (J) â (J) ( J) ĉ = - 4 K â J This is the K 4 of theory (3) eqution (8) n the ompenstion we get :- 09

4 Tikrit Journl of Pure Siene () 06 ISSN: = - 4[ h h δ + δ δ )] = - 8 h h - (n ) δ (n ) δ δ + δ δ ).(5) From eqution () n (5) it follows λ(, )= λ(, ) = - 8 h h - δ + δ δ ) Thus y efinition (6) we get : M is onstnt type if n only if λ(, )= -8 h h - (n ) δ (n ) δ + δ δ )+ + δ δ )+ + δ δ )+ + δ δ )+ δ + δ δ ) Now we stuing Nerly Khler mnifol of pointwise holomorphi setionl onhormoni (PMK M ()) urvture tensor. Let {M n,g,j} Almost Hermitin mnifol (M) moule smooth vetor fiel on mnifol M n Definition (8): [ ] Two imensions Level two-imensionl σ T m (M),m M lle holomorphi, if J m (σ)= σ Theorem (9) :[ ] Two imensions Level two-imensionl σ T m (M),m M holomorphi if n only if when σ = L(, J)where (M)- some vetor, L enote tking liner hull Defintion (0):[3 ] Setionl urvture lmost Hermitin mnifol {M,g,J} on the iretion two imensions Level two-imensionl σ T m (M),m M lle holomorphi setionl urvture (on t equl zero ) vetor σ n enote H m (),With form H m () = <Rm (,J)J,)> 4 ; m M, T m (M) - If H m () Do not epen on the hoie of the point σ T m (M) then mnifol involving M pointwise onstnt holomorphi setionl urvture. - If Hm()Do not epen on the hoie of the point m then mnifol involving M glool onstnt holomorphi setionl urvture. Definition () : Let M e n lmost Hermitin mnifol, holomorphi setionl urvture onhormoni (HK m ()- urture) tensor of mnifol M in the iretion (M), 0 is funtion H() whih is efin s : K (, J,,J ) = MK m () 4, (M) (HK m () = = <Km (,J)J,)> 4 ; m M, T m (M) where = <,> (6) Definition (): A mnifol M is lle mnifol pointwise onstnt holomorphi setionl onhormoi urvture (PHK m ()) urvture tensor, if HK m () oes not epen on i.e K (, J,,J ) = 4 ; (M), C (M).(7) Definition (3): A mnifol M pointwise onstnt holomorphi setionl onhormoi urvture is lle glol onstnt holomorphi setionl onhormoi urvture if HK m () hs onstnt i.e (HK m () oes not epen on m M ) Theorem(4): [5] Almost Hermitin mnifol (M,g,J) is mnifol onstnt HK- urvture if n only if,,when in the joint G-struture spe δ = K Lemm(5): [4] If M lmost Hermitin mnifol of pointwise holomorphi setionl urvture tensor then we hve: 4 = δ Theorem (6) : Nerly Khler mnifol of pointwise onstnt holomorphi setionl onhrmoni (PHK m ()) urvture ) urvture tensor if the omponents of holomorphi setionl (HS-urvture) urvture tensor in the joint G-struture spe tht stisfies the following onition: = δ A Proof : Suppose tht M is Nerly Khler mnifol of PHKurvtue tensor Aoring to the the efinition () we hve : K (, J,,J ) = 4.(8) Where C (M) y using Lemm(5),the eqution (8) eomes:- K (, J,,J ) = δ.(9) In the joint G-struture spe the eqution (9) n e written s the following form: K ijkl i j () k l = K ( J) ĉ ( J) ( J) ĉ ĉ +K ĉ +K â ĉ ĉ ĉ â â â â â ĉ â ĉ â + K â = δ y using the properties =, â = â we get :- 0

5 Tikrit Journl of Pure Siene () 06 ISSN: < K(, J), J, > = - K â K â â + 4 K + K â ĉ + K â ĉ K = δ y using the properties of onhrmoni tenser eqution (5) We get : K â K â T.e K Similrly: K, K we get : = K â =, K, K Referenes. Aul - Qer R.T., "On the geometry of onhrmoni urvture tensor of Khler n nerly Khler mnifols", M.S. thesis, University of Tikrit, College of Eution for Pure Sienes, 05.. Kirihenko V.F. "Differentil - Geometril struture on mnifol", Mosow Stte Pegogil University, Kirihenko V.F. n Vlsov L.I., Conirulr gemetry of nerly Khler mnifol,sornik: mthemtis, V.93,No.5,pp ,00. < K(, J), J, > = 4 K â = δ y theorm (3) the eqution (8 ) we show : 4 {A h δ δ (n ) + δ δ } = δ If we ompute nti symmetrizing struture tensor we hve : A δ + δ δ + δ δ + δ δ ) = δ 4(n ) h Explin tht A tensor type ( ) with omponent 0 0 A = A δ + δ δ + δ δ ) 4(n ) + δ δ Symmetri for eh Ientil for eh pir supernl or sujent n we get : A = δ. 4. Sn M.J., "Complex onirulr urvture tensor of ertin lsses of lmost Hermition mnifol", (Mster thesis). University of srh, Shih A.A., "Geometry of the tensor of onhrmoni urvture of nerly Khlerin mnifols", (PhD thesis). Mosow Stte Pegogil Univ., (in Russin), Shih A.A., Kirihenko V.F, Rustnoy A.R, "On geometry of the tensor of onhrmoni urvtur of lmost Hermitin",Mthemtil Notes, Mosow, Vol.90, No-, pp , 0. الملخص النوع الثابت )K( لمنطويات كوهلر ومنطوي كوهلر التقريبي لتنزر االنحناء الكونهورموني علي عبد المجيد شهاب رنا حازم جاسم قسم الرياضيات كلية التربية للعلوم الصرفة جامعة تكريت تكريت الع ارق ان الشروط الثابتة لألنواع الكونهورمونية لمنطوي كوهلر ومنطوي كوهلر التقريبي نحصل عليها عندما يكون منطوي كوهلر التقريبي هو منطوي كونهورموني من النوع الثابت )K(. وكذلك اثبات ان )M( هو منطوي كوهلر تقريبي ذات ثابت نقطي هلومورفك في القاطع الكونهورموني ) urvture (PHKm()) لتنزر االنحناء اذا كانت محتويات القاطع الهولومورفك urvture) (HS- لتنزر االنحناء في بنية الفضاء المركب G يحقق الشروط محدد.