Aab. J. Math. 207 6:35 327 DOI 0.007/s40065-07-072-6 Aabia Joua of Mathematics E. Peygha L. Noumohammadifa A. Tayebi,-Teso sphee bude of Cheege Gomo type Received: 3 Jauay 207 / Accepted: 30 Api 207 / Pubished oie: 6 May 207 The Authos 207. This atice is a ope access pubicatio Abstact We costuct a metica famed f 3, -stuctue o the, -teso bude of a Riemaia maifod equipped with a Cheege Gomo type metic ad by estictig this stuctue to the, -teso sphee bude, we obtai a amost metica paacotact stuctue o the, -teso sphee bude. Moeove, we show that the, -teso sphee budes edowed with the iduced metic ae eve space foms. Mathematics Subect Cassificatio 53C5 53C2 Itoductio Maybe, the best ow Riemaia metic o the taget bude is itoduced by Sasai i 958 [20]. Howeve, i most cases, the study of some geometic popeties of the taget bude equipped with this metic ead to the fatess of the base maifod. A few yeas ate, some eseaches became iteested i fidig othe ifted stuctues o the taget budes, cotaget, ad taget sphee budes with iteestig popeties see [2,4 0,3,6,2]. The taget sphee bude T M cosistig of sphees with costat adius see as hypesufaces of the taget bude TM has sigificat appicatios i geomety [,2]. Recety, some iteestig esuts wee obtaied by edowig the taget sphee budes with Riemaia metics iduced by the atua ifted metics fom TM, which ae diffeet fom Sasaia see [,8,5]. Teso budes Tq p M of type p, q ove a diffeetiabe maifod M ae pime exampes of fibe budes, which ae studied by mathematicias such as Ledge, Yao, Cegiz, ad Saimov [3,4,8]. The tagetbude TMad cotaget bude T M ae the specia cases of Tq p M. E. Peygha B L. Noumohammadifa Depatmet of Mathematics, Facuty of Sciece, Aa Uivesity, Aa, Ia E-mai: e-peygha@aau.ac.i A. Tayebi Depatmet of Mathematics, Facuty of Sciece, Qom Uivesity, Qom, Ia E-mai: aba.tayebi@gmai.com
36 Aab. J. Math. 207 6:35 327 Saimov ad Geze [9] itoduced the Sasai metic S g o the, -teso bude T M of a Riemaia maifod M ad studied some geometic popeties of this metic. By the simia method used i the taget bude, the peset authos defied i [7] the Cheege Gomo type metic CG g o T M which is a extesio of Sasai metic. The, the authos studied some eatios betwee the geometic popeties of thebasemaifodm, g ad T M, CG g. I the peset pape, we coside Cheege Gomo type metic CG g o T M, ad by appyig it, we itoduce a metica famed f 3, -stuctue o T M. The, by estictig this stuctue to the, -teso sphee bude of costat adius, T M, we obtai a metica amost paacotact stuctue o T M. Fiay, we show that the, -teso sphee budes edowed with the iduced metic ae eve space foms. 2 Peimiaies Let M be a smooth -dimesioa maifod. We defie the bude of, -teos o M as T M = p M T p, whee deotes the disoit uio, ad we ca it, -teso bude. We aso defie the poectio π : T M M to p. Ifxi ae ay oca coodiates o U M, adp U, the coodiate vectos { i,whee i :=, fom a basis fo T x i p M whose dua basis is dx i. Ay teso t T M ca be expessed i tems of this basis as t = t i i dx. Fo ay coodiate chat U,x i o M, coespodece t T x x,ti U R2 detemies oca tiviaizatios φ : π U T M U R2, which shows that T M is a vecto bude o M. Theefoe, each oca coodiate eighbohood {U, x = i M iduces o T M a oca coodiate eighbohood {π U; x, x = t i =, = +, i.e., T M is a smooth maifod of dimesio + 2. We deote by FM ad I M, the ig of ea-vaued C fuctios ad the space of a C teso fieds of type, o M. Ifα I M, the by cotactio, it is egaded as a fuctio o T M,whichwe deote by ıα.ifα has the oca expessio α = α i dx i i a coodiate eighbohood Ux M,the x ıα = αt has the oca expessio ıα = α i ti with espect to the coodiates x, x i π U. Suppose that A I M. The, the vetica ift V A I 0 T M of A has the foowig oca expessio with espect to the coodiates x, x i T M: V A = V A, 2. whee V A = A i ad := =. Moeove, if V I x t i 0 M, the the compete ift C V ad the hoizota ift H V I 0 T M of V to T M have the foowig oca expessios with espect to the coodiates x, x i T M see [3]ad[4]: C V = V + t m H V = V + V s Ɣ m s ti m Ɣi sm tm m V i tm i V m, 2.2, 2.3 whee Ɣi ae the oca compoets of a symmetic affie coectio o M. Let Ux h be a oca chat of M. Usig 2.ad2.3, we obtai e := H = H δ h h = δ h h + Ɣ s h t s Ɣ s ts h h, 2.4 e := V i dx = V δ i δ h dx h = δ i δ h h, 2.5 whee δ h is the Koece s symbo ad = +,..., + 2.These + 2 vecto fieds ae ieay idepedet ad geeate the hoizota distibutio of ad vetica distibutio of T M, espectivey. Ideed,wehave H X = X e ad V A = A i e see [9]. The set {e β={e, e is caed the fame adapted to the affie coectio o π U T M.
Aab. J. Math. 207 6:35 327 37 Lemma 2. Let α, α 2, α 3, ad α 4 be smooth fuctios o T M, such that α g ti g δ m δv + α 2g i g m δ δv t + α 3 t m t i δ δv t + α 4 t t t i δm δv = 0. 2.6 The, α = α 2 = α 3 = α 4 = 0. Poof Cotactig 2.6 with t v, the diffeetiatig the obtaied expessio thee times, it foows that, α 3 = α 4. Aso diffeetiatig the emaiig expessio two times, we have α g ti g t m α 2g i g m t t = 0. Cotactig the above equatio with t i, yied α = α 2. Mutipyig 2.6 byg h g i ad δm h δ, we obtai α 3 = α 4 = 0. Fiay cotactig 2.6 with t i, tm, we cocude that α = α 2 = 0. 3 Cheege Gomo type metic o T M Fo each p M, the extesio of the scaa poduct g, deoted by G, is defied o the teso space π p = T p by GA, B = gitg A i Bt, A, B I p, whee g i ad g i ae the oca covaiat ad cotavaiat tesos associated with the metic g o M. Now, we coside o T M a Riemaia metic CG g of Cheege Gomo type, as foows [7]: CG g V A, V B = V aga, B + bgt, AGt, B, CG g H X, H Y = V gx, Y, CG g V A, H Y = 0, 3. fo each X, Y I 0 M ad A, B I M, wheea ad b ae smooth fuctios of τ = t 2 = t i tt g itxg x o T M that satisfies the coditios a > 0ada + bτ >0. The symmetic matix of type 2 2 g 0 0 ag g it + b t, 3.2 i t t associated with the metic CG g i the adapted fame {e β,hastheivese g 0 0 a g g it aa+bτ b, 3.3 ti tt whee t i = g h g i th. I the specia case, if a = adb = 0, we have the Sasai metic S g see [9]. Let ϕ = ϕ i dx be a teso fied o M. The, γϕ = t m x i ϕi m ad γϕ = t i x m ϕm ae vecto x fieds o T M. The bacet opeatio of vetica ad hoizota vecto fieds is give by the fomuas [ V A, V B]=0, [ H X, V A]= V X A, 3.4 [ H X, H Y ]= H [X, Y ]+ γ γrx, Y, 3.5 whee R deotes the cuvatue teso fied of the coectio ad γ γ : ϕ I 0 T M is the opeato defied by 0 γ γϕ = t m ϕi, ϕ I M. m t i m ϕm
38 Aab. J. Math. 207 6:35 327 Popositio 3. [7] The Levi-Civita coectio CG associated with the Riemaia metic CG gothe, -teso bude T M has the fom CG e e = Ɣ e + R s 2 tv s R s v ts e, s CG e e = a 2 CG e e = a 2 CG e e = g ta R s ts a gb Rts ts b g ia R s ts a g b Ris ts b e, e + Ɣi v δ Ɣ Lt t δ δi v + t i δ δv t + Mg g ti t v + Nt t t i tv whee R ae the compoets of the cuvatue teso fied of the Levi-Civita coectio o the base maifod M, g ad L := a a,m:= a +2b a+bτ, ad N := b a 2a b aa+bτ. I the foowig sectios, we coside the subset T M of T M cosistig of sphee of costat adius.now, we coside the, -teso fied P o T M as foows: [7] P H X = c V X Ẽ + d gx, E V E Ẽ, P V X Ẽ = c H 2 X + d 2 gx, E H E, P V A = V A, whee c, c 2, d,add 2 ae smooth fuctios of the eegy desity t ad Ẽ = g E I 0 M. Usig the adapted fame {e i, E e, e to T M, P has the foowig ocay expessio: Pe i = c E e + d E i E v E e, PE e = c 2e i + d 2 E i E e, 3.6 Pe = e, whee E = g E.Wehave Theoem 3.2 [7] The atua teso fied P of type, o T M, defied by the eatios 3.6,isaamost poduct stuctue o T M, if ad oy if its coefficiets ae eated by δv i e, e, c c 2 =, c + d E 2 c 2 + d 2 E 2 =. 3.7 Theoem 3.3 [7] CG g, P is a Riemaia amost poduct stuctue o T M if ad oy if ad 3.7 hod good. c = Now, we coside vecto fieds ad -foms a E, c 2 = E a, d = 2 a E 3, d 2 = 2 a E, 3.8 ξ := α H E, ξ 2 := β V E Ẽ, ξ 3 := κ V A, 3.9 η = γ E v dx v, η 2 = λe v E δt v, η3 = ρ t v δtv, 3.0 o T M,wheeα, β, κ, γ, λ, adρ ae smooth fuctios of the eegy desity o T M ad δtv is a dua of e. Usig 3.6ad3.9, we get Pξ = α β c + d E 2 ξ 2, Pξ 2 = β α c 2 + d 2 E 2 ξ, Pξ 3 = ξ 3, 3. ad η ξ = αγ E 2, η 2 ξ 2 = βλ E 4, η 3 ξ 3 = κρτ, η a ξ b = 0, 3.2
Aab. J. Math. 207 6:35 327 39 whee a, b =, 2, 3 with coditio a = b. We have aso the foowig equatios usig 3.6ad3.0: η P = γ λ E 2 c 2 + d 2 E 2 η 2, η 2 P = λ E 2 c + d E 2 η, η 3 P = η 3. 3.3 γ Now, we defie a teso fied p of type, o T M by px = PX η Xξ 2 η 2 Xξ η 3 Xξ 3. 3.4 This ca be witte i a moe compact fom as p = P η ξ 2 η 2 ξ η 3 ξ 3.Fom3.4, the foowig oca expessio of p yieds: pe i = c δ i v + d βγe i E v E e, pe e c = 2 δi + d 2 αλ E 2 E i E e, 3.5 pe = δ δi v κρ t i tv e. Lemma 3.4 We have pξ = β α c + d βγ E 2 ξ 2, pξ 2 = β α c 2 + d 2 αλ E 2 E 2 ξ, pξ 3 = κρτξ 3, η p = γ c λ E 2 2 + d 2 αλ E 2 E 2 η 2, η 2 p = λ E 2 γ c + d βγ E 2 η, η 3 p = κρτ η 3, 3.6 3.7 β p 2 = I α c 2 + d 2 E 2 + λ E 2 c + d E 2 βλ E 4 η ξ γ α β c + d E 2 + γ λ E 2 c 2 + d 2 E 2 αγ E 2 η 2 ξ 2, + κρτ 2η 3 ξ 3. 3.8 Poof We oy pove 3.8. Usig 3., 3.2, ad 3.3, we have p 2 X = ppx = P [ PX η Xξ 2 η 2 Xξ η 3 ] Xξ 3 η [ PX η 2 ] Xξ ξ2 η 2 [ PX η ] Xξ 2 ξ η 3 [ PX η 3 ] Xξ 3 ξ = X β c2 + d 2 E 2 η Xξ α α c + d E 2 η 2 Xξ 2 γ c2 β λ E 2 + d 2 E 2 η 2 Xξ 2 The above equatio gives us 3.8. Lemma 3.5 Let P satisfy Theoem 3.2.If + E 2 αγη 2 Xξ 2 2η 3 Xξ 3 λ E 2 γ + E 4 βλη Xξ + κρτη 3 Xξ 3. c + d E 2 η Xξ αγ E 2 =, βλ E 4 =, κρτ =, λ = γ E 2 c 2 + d 2 E 2, 3.9 the p 3 p = 0 ad p has the a + 2 3 o coa 3.
320 Aab. J. Math. 207 6:35 327 Poof If 3.9 hods, the fom the above emma, we obtai p 2 = I η ξ η 2 ξ 2 η 3 ξ 3, pξ = 0, η ξ = δ, η p = 0, 3.20 whee, =, 2, 3. Theefoe, we have p 3 = p. To pove the secod pat of the emma, it is sufficiet to show that e p = spa{ξ,ξ 2,ξ 3. Fom the secod eatio i 3.20, we otice that spa{ξ,ξ 2,ξ 3 e p. Now, et X = X e + X v E e + X e e p. The, px = 0 impies that Thus Sice P 2 = I, the usig 3., we get PX η Xξ 2 η 2 Xξ η 3 ξ 3 = 0. P 2 X = η XPξ 2 + η 2 XPξ + η 3 XPξ 3. X = β α c 2 + d 2 E 2 η Xξ + α β c + d E 2 η 2 Xξ 2 + η 3 Xξ 3, that is X spa{ξ,ξ 2,ξ 3, i.e., e p spa{ξ,ξ 2,ξ 3. Theoem 3.6 Let P be the amost poduct stuctue chaacteized i Theoem 3.2 ad ξ, η, =, 2, 3, ad p be defied by 3.9, 3.0, ad 3.4, espectivey. The, the tipe p,ξ, η povides a famed f 3, - stuctue if ad oy if 3.9 hods. Poof Let p,ξ, η be a famed f 3, -stuctue o T M. The, by the defiitio of a famed f 3, -stuctue, we have η ξ = δ,whee, =, 2, 3. Thus, 3.2givesus αγ E 2 = βλ E 4 = κρτ =. 3.2 We have aso pξ 3 = 0. The above equatio ad the secod eatio i 3.6 yied λ = γ c E 2 2 + d 2 E 2. Usig Lemmas 3.4 ad 3.5, the covese of the theoem is poved. Lemma 3.7 Let CG g, P satisfy Theoem 3.3. The, the Riemaia metic CG g satisfies CG gpx, py = CG 2c + d E 2 gx, Y aβ β E 2 E 2 η Xη Y γ 2c2 + d 2 E 2 α η 2 Xη 2 Y fo each X, Y I 0 T M. κa + bτ λ E 2 α E 2 2 ρ κτ η 3 Xη 3 Y, Poof Obviousy, we have CG gξ,ξ 2 = 0. Usig 3.9, we deduce We have aso CG gξ,ξ = α 2 E 2, CG gξ 2,ξ 2 = aβ 2 E 4, CG gξ 3,ξ 3 = κ 2 a + bττ. CG gx,ξ = α γ η X, CG gx,ξ 2 = aβ λ η2 X, CG gx,ξ 3 = κ ρ a + bτη3 X. Usig 3.3 ad the above equatios, we deduce CG gpx, py = CG gpx, PY 2aβ γ c + d E 2 E 2 η Xη Y + α 2 E 2 η 2 Xη 2 Y + aβ 2 E 4 η Xη Y 2α c2 λ E 2 + d 2 E 2 η 2 Xη 2 Y
Aab. J. Math. 207 6:35 327 32 2 κa + bτ ρ κτ η 3 Xη 3 Y. Howeve, CG gpx, PY = CG gx, Y, sice CG g, P is a Riemaia amost poduct stuctue. Thus, the emma is poved. Theoem 3.8 If CG g, P is the Riemaia amost poduct stuctue chaacteized i Theoem 3.3, ad ξ, η,=, 2, 3, p ae defied by 3.9, 3.0, ad 3.4, espectivey, the CG g, p,ξ, η povides a metica famed f 3, -stuctue if ad oy if 3.9 ad hod good. Poof Usig Lemma 3.7, it is easy to see that the meticity coditio γ = α, λ = aβ, ρ = κa + bτ, 3.22 CG gpx, py = CG gx, Y η Xη Y η 2 Xη 2 Y η 3 Xη 3 Y, of the famed f 3, stuctue chaacteized by 3.9issatisfiedifadoyif3.22 hods good. Thus, the poof is compete. 4O, -teso sphee bude Let be a positive umbe. The, the, -teso sphee bude of adius ove a Riemaia M, g is the hypesuface T M ={x, t T M G xt, t = 2. It is easy to chec that the teso fied N = t i e, is a teso fied o TM which is oma to T M. I geea fo ay teso fied A I M, the vetica ift V A is ot taget to T M at poit x, t. We defie the tagetia ift T A of a teso fied A to x, t T M by T A x,t = V A x,t 2 G xa, tn x,t. 4. Now, the taget space TT M is spaed by e ad e T = t 2 i tv. We otice that thee is the eatio t i et = 0, ad hece, i ay poit of T M, the vectos et, = +,..., + 2,spaa 2 - dimesioa subspace of TT M. Usig 4. ad the computatio statig with the fomua 3., we see that the Riemaia metic g o T M, iduced fom CG g, is competey detemied by the idetities g T A, T B = a V GA, B Gt, AGt, B, 2 g T A, H Y = 0, 4.2 g H X, H Y = V gx, Y, fo a X, Y I 0 M ad A, B I M, wheea is costat that satisfy a > 0. The bacet opeatio of tagetia ad hoizota vecto fieds is give by the fomuas [ ] e T, e T = 2 t t δv i δ t i δv t δ [ ] e, e T = Ɣi v δ Ɣ δv i e T, [ ] e, e = R s tv s R s v ts e T, e T. Usig the Levi-Civita coectio of the Cheege Gomo type metic itoduced by the authos i [7], we ca cocude the foowig:
322 Aab. J. Math. 207 6:35 327 Popositio 4. The Levi-Civita coectio, associated with the Riemaia metic g o the teso bude T M, has the fom e e = Ɣ e + R s 2 tv s R s v ts e T, e = a g e T ta R s ts a 2 gb Rts ts b e, et e = a g ia R s ts a 2 g b Ris ts b e + Ɣi v δ Ɣ δv i e T, et e T = 2 t i δ δv t et. 4. A amost paacotact stuctue o T M I this sectio, we show that the famed f 3, -stuctue o T M, give by Theoem 3.6, iduces a amost paacotact stuctue o T M. Fist, we show that ξ 2 ad ξ 3 ae uit oma vecto fieds with espect to the metic CG g.let x i = x i u α, t i = ti uα, α {,...,, 4.3 be the oca equatios of T M i T M.Siceτ = ti tt g g it = 2,wehave Howeve, we have τ x = 2 Ɣ s ts h Ɣs h t s t h, τ th By epacig 4.5ito4.4, we get x Ɣ s ts h Ɣs h t s u α + t h u α t h The atua fame fied o T M is epeseted by u α = x u α x + t h u α τ x x u α + τ th th = 0. 4.4 uα t h = x u α e + = 2 t h. 4.5 = 0. 4.6 x Ɣ s ts h Ɣs h t s u α + t h u α e h. 4.7 The, by 4.6, we deduce that x CG g u α,ξ 3 = κa + bτ Ɣ s ts h Ɣs h t s u α + t h u α t h = 0. 4.8 Simiay, we obtai CG g u α,ξ 2 = 0. Thus, ξ 2 ad ξ 3 ae othogoa to ay vecto taget to T M.The vecto fied ξ is taget to T M,siceCG gξ,ξ 2 = 0. Lemma 4.2 O T M, we have η 2 = η 3 = 0, px = PX η Xξ, X χt M. Poof Usig η i T M X = CG gx,ξ i = 0, i = 2, 3, the poof is obvious. We put ξ T M = ξ, η T M = η ad p T M = p. The, Theoem 3.6 ad Lemma 4.2 impy the foowig.
Aab. J. Math. 207 6:35 327 323 Theoem 4.3 If 3.9 hods, the the tipe p,ξ,ηdefies a amost paacotact stuctue o T M, that is, i ηξ =, pξ = 0, η p = 0. ii p 2 X = X ηxξ, X χt M. It is easy to show that if 3.9ad3.22 hod, the the Riemaia metic g satisfies gpx, py = gx, Y ηxηy, X, Y χt M. 4.9 Usig the equatio 4.9 ad Theoem 4.3, we cocude the foowig: Theoem 4.4 If 3.9 ad 3.22 hod, the the esembe p,ξ,η, g defies a amost metica paacotact stuctue o the taget sphee bude T M. 4.2 No-existece, -teso sphee budes space fom The cuvatue teso fied R of the coectio is defied by the we-ow fomua R X, Ỹ Z = X Ỹ Z Ỹ X Z [ X,Ỹ ] Z, whee X, Ỹ, Z I 0 T M. Usig the above equatio, Popositio 4., ad the oca fame {e, e,we T obtai Re m, e e = HHHHm e + HHHT met, 4.0 Re m, e e T = HHTH m e + HHTT m et, 4. Re m, e T e = HTHH m e + HTHT m et, 4.2 Re m, e T e T = HTTH m e, 4.3 Re T m, e et = TTHH m e, 4.4 Re T m, e et T = TTTT m et, 4.5 whee HHHHm = Rm + a 4 {g a R sh m R p h + g a R sh R mp Rsh Rsh R p mh 2Rsh m R p + 2Rsh Rmp R p mh t a s t p h t a s t p + g hb Rp R mh s R pm R h s + 2R p Rmh s t p b t s + g hb Rsm R p R s Rmp 2R s Rmp tb s t p h, HHHTm = { m R s 2 tv s Rm s tv s + Rms v ts m Rs v ts, HHTH = a { g m ia m R s ts a 2 g ia R s m ts a + g b Rism ts b g b m R HHTT = R v m mi δ R m δv i + a {g ia Rmh s 4 R p h Rh s R p m h ts v ta p + g ia Rhp v Rs m h Rmhp v Rs h ts a t p + g b Rh s R ipm h R + g b Rmhs v R ip h Rhs v Ripm h t s t p b + R s 2 m tv s R HTHH m = a { g ta m R s ts a 2 gb m Rts ts b, HTHT = R m m 2 δv t Rmt v δ + a { g ta R p h s Rmh 4 tv s ta p is ts b, mh s R ip h ms v ts t i, t p b tv s
324 Aab. J. Math. 207 6:35 327 HTTH TTHH TTTT m m m g b R + g b R h tp h R s mh tv s t p b g tar s h tp R mhs v ts t p b, = a g Ritm 2 g it R m + a2 4 g ta R s h g ibr p h g a Rtsh g b R g b R tsm ts b t i, m ts a tb p + gb R ipm h ts a t p b = a g t R m g m Rt g ta R s h g br pm h g a R sm h g b R h ph g tar s h g m g i δ m δv + a2 R 4 ts a tb p + g tar s v mhp t p ts a { g ta R s h g b R tph g iar s m h t p b ta s a 2 2 g ta R s m ta s { g a R sm ipm h ta s t p b h g tbr p h ts a tb p h gmb Rp h ta s t p b t p b ta s sh gb Rts h t p b ts a tp ta s t p b + gb Rtph g a R sm h g mb R t p b ta s + gma R g a Rtsh gmb Rp h t p b ts a, = 4 t m t i δ δv t t t t i δm δv. + 2 g g ti δ m δv I the foowig, we cacuate the Ricci teso Ric of T M, g usig the we-ow fomua: Ric = tacex R X, Ỹ Z, X, Ỹ, Z I 0 T M. Let E,...,E 2 + be the othooma fame, such that the fist vectos E,...,E ae vectos of a fame i HTM ad the ast 2 vectos E +,...,E 2 + ae vectos of a fame i VTM [8]. We coside the ast vecto E 2 + as the uitay vecto of the oma vecto N = t i e to T M. It is easy to see that the vecto fieds e T,...,eT ae ot idepedet. Cosideig the basis e 2,...,e, e T,...,eT 2 fo TT M, oa ope set of T M whee ti = 0, we ca wite the ast vecto et as foows: 2 e T 2 = e T = t Usig the defiitio of the Ricci teso, we have i, = i = = t i et. Rice T, e T = TTTT + HTTH. Diect cacuatios give us TTTT s e T = =,h= =h =,h= =h = = TTTT TTTT TTTT s s s e T + TTTT e T TTTT e T TTTT s t e T s 2 s t t v et.,h= =h = t h et
Aab. J. Math. 207 6:35 327 325 Settig s = i the above equatio, we have TTTT = TTTT t t v TTTT Note that i the eft side of the above equatio, summatio idex is diffeet fom the summatio idex i the ight side. Usig the above expessio of TTTT ad 4.5, we get Hece It foows that: Ric e T, e T t v TTTT t v TTTT t = 2 2 2 m = 2 g g ti t 4 t t t i t. = 2 g g ti 4 t t t i. = TTTT + HTTH 2 g g ti + 4 t t t i g ti g 2 t i t t + a2 { g b R 4 g ta R s h g ibr p h ts a tb p ga Rtsh g b Rip h ts a t p b + g ta R s h g b R h. ip ta s t p b. tph g iar s h t p b ta s I a simia way, we get othe compoets of the Ricci teso o T M as foows: Rice T, e = HTHH = a { g ta R s ts a 2 gb Rts ts b, Rice, e T = HHTH = a { g ia R s ts a 2 g b Ris ts b, Rice, e = HHHH + THHT t t v THHT = R + a { g hb Rp Rh s 2 t p b t s g ar sh R p h ta s t p g hb Rs Rp ts b t p h + g ar sh Rp ta s t p h a { g a R sh R p h 4 ta s t p + g va R p h Rh s tv s ta p +g hb Rs R p ts b t p h + gb Rvp h R v. hs ts t p b Theoem 4.5, -teso sphee bude T M, with the Riemaia metic g iduced fom the metic CG g o T M, has eve costat sectioa cuvatue. Poof It is ow that the cuvatue teso fied of the Riemaia maifod T M, g with costat sectio cuvatue satisfies the eatio R X, Ỹ Z = { gỹ, Z X g X, ZỸ, 4.6 whee X, Ỹ, Z I 0 T M. IfT M, g has costat sectioa cuvatue, thewehave { R e T m, e et T ge T, e T et m g e T m, e et T = 0. 4.7 Usig 4.7ad4.5, we get 2 a 2 [ g ti g δ m δv g ig m δ δv t + 2 t m t i δ δv t t t t i δm δv ] = 0. 4.8
326 Aab. J. Math. 207 6:35 327 Usig the above equatio ad Lemma 2., we deduce = 0ada =.SiceT 2 M, g has costat sectioa cuvatue, wehave Re m, e e { ge, e e m ge m, e e = 0. 4.9 4.0ad4.9giveus Rm g δm g mδ a + {g a R sh m 4 R p h R sh R p mh 2Rsh R p mh ts a t p + g a R sh Rmp R sh m R p + 2Rsh Rmp ts a t p h + ghb Rsm R p Rs Rmp 2R s Rmp tb s t p h + ghb Rp R mh s Rpm R h s + 2R p Rmh s t p b t s = 0. 4.20 Diffeetiatig the expessio 4.20 two times, i the tagetia coodiates x ; =,..., + 2,we cocude Rm = g δm g mδ. 4.2 I additio, we have { R e T m, e e T g e, e T e T m g e T m, e et = 0. 4.22 Settig a = ad 4.2i4.3 ad the usig 4.22, we obtai 2 [ 2 2 g g tm δi g im δt + 2g it δm ] + git g δm g δm [g 4 4 ta g b g pm g s δi ta s t p b g img s ts a t b g pmg ti a t p b + g img t a p t p b + g ta g ib g s g ts a tb m gs g p δmt s a tb p + g g sp δmt s a tb p g g s ts a tb m + g a g b g sp g im δt ts a t p b g sig pm δt ts a t p b + g tig pm ta t p b g tpg im ta t p b + g ia g b δmδ t t p b ta p δ t t b ta m δ mt b ta t + δt tb ta m + [ 2 4 g ta g s δm ta s g tag tm a g smg b δt ts b + g tmg b tb + 2δ m t t Fom the above equatio i the poit x i, t i = xi,δ i T M,weget 2 2 [ g g tm δ i g im δ t + 2g it δ m ] + git g δ m g δ m ] t i = 0. ] + 4 δ m δ t δ i = 0, which is a cotadictio. Thus, we cocude that the maifod T M, g may eve be a space fom. Fo Sasai metic S g we have a =. The usig Theoem 4.5,wehave Cooay 4.6 The, -teso sphee bude T M, edowed with the metic iduced by the Sasai metic S gfomt M, is eve a space fom. I this pape, we show that cosideig Cheege Gomo type metic CG g o T M, we ca costuct a metica famed f 3, -stuctue o T M. I additio, by estictig this stuctue to the, -teso sphee bude with costat adius, T M, we obtai a metica amost paacotact stuctue o T M. Moeove, we deduce that, -teso sphee budes edowed with the iduced metic ae eve space foms. Ope Access This atice is distibuted ude the tems of the Ceative Commos Attibutio 4.0 Iteatioa Licese http:// ceativecommos.og/iceses/by/4.0/, which pemits uesticted use, distibutio, ad epoductio i ay medium, povided you give appopiate cedit to the oigia authos ad the souce, povide a i to the Ceative Commos icese, ad idicate if chages wee made.
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