ON THE GEOMETRY OF FRAME BUNDLES. Kamil Niedziałomski

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1 ARCHIVUM MATHEMATICUM BRNO) Toms 8 01), ON THE GEOMETRY OF FRAME BUNDLES Kaml Nedzałomsk Abstract. Let M, g) be a Remannan manfold, LM) ts frame bndle. We constrct new examples of Remannan metrcs, whch are obtaned from Remannan metrcs on the tangent bndle T M. We compte the Lev Cvta connecton and crvatres of these metrcs. 1. Introdcton Let M, g) be a Remannan manfold, LM) ts frame bndle. The frst example of a Remannan metrc on LM) was consdered by Mok [1]. Ths metrc, called the Sasak Mok metrc or the dagonal lft g d of g, was also nvestgated n [5] and [6]. It s very rgd, for example, LM), g d ) s never locally symmetrc nless M, g) s locally Ecldean. Moreover, wth respect to the Sasak Mok metrc vertcal and horzontal dstrbtons are orthogonal. A wder and less rgd class of metrcs ḡ, n whch vertcal and horzontal dstrbtons are no longer orthogonal, has been recently consdered by Kowalsk and Sekzawa n the seres of papers [9, 10, 11]. These metrcs are defned wth respect to the decomposton of the vertcal dstrbton V nto n = dm M sbdstrbtons V 1,..., V n. In ths short paper we ntrodce a new class of Remannan metrcs on the frame bndle. We dentfy dstrbtons V wth the vertcal dstrbton n the second tangent bndle T T M. Namely, each map R : LM) T M, R 1,..., n ) = ndces a lnear somorphsm R : H V T T M, where H s a horzontal dstrbton defned by the Lev Cvta connecton on M. By ths dentfcaton we pll back the Remannan metrc from T M. We pll back natral metrcs, n the sense of Kowalsk and Sekzawa [8], from T M and stdy the geometry of sch Remannan manfolds. We compte the Lev Cvta connecton, the crvatre tensor, sectonal and scalar crvatre.. Remannan metrcs on frame bndles Let M, g) be a Remannan manfold. Its frame bndle LM) conssts of pars x, ) where x = π LM) ) M and = 1,..., n ) s a bass of a tangent space 010 Mathematcs Sbject Classfcaton: prmary 53C10; secondary 53C, 53A30. Key words and phrases: Remannan manfold, frame bndle, tangent bndle, natral metrc. The athor s spported by the Polsh NSC grant No. 6065/B/H03/011/0. Receved May, 01, revsed May 01. Edtor O. Kowalsk. DOI: /AM

2 198 K. NIEDZIAŁOMSKI T x M. We wll wrte nstead of x, ). Let x 1,..., x n ) be a local coordnate system on M. Then, for every = 1,..., n, we have = j for some smooth fnctons j on LM). Pttng α = x π LM), α, j k ) s a local coordnate system on LM). Let ω be a connecton form of LM) correspondng to Lev Cvta connecton on M. We have a decomposton of the tangent bndle T LM) nto the horzontal and vertcal dstrbton: j T LM) = H LM) x j V LM), where H LM) = ker ω and V LM) = ker π LM). Let X h denotes the horzontal lft of a vector X T x M, π LM) ) = x, to H LM). Let L : GLn) LM), L a) = a, be a left mltplcaton of a GLn) by a bass LM). Let A = L A) be a fndamental vertcal vector correspondng to a matrx A gln). Denote by V a lnear sbspace of V LM) spanned by fndamental vertcal vectors A, where the matrx A gln) has only nonzero -th colmn. For an ndex = 1,..., n defne a map R : LM) T M as follows R ) =, = 1,..., n ) LM). R s the rght mltplcaton by a -th vector of a canoncal bass n R n. We wll need some basc facts abot the second tangent bndle T T M. There s a decomposton of T T M nto horzontal and vertcal part, T ζ T M = Hζ T M Vζ T M, wth respect to the connecton map K : T T M T M and the projecton n the tangent bndle π T M : T M M, see for example [7]. Let X h,t M ζ and X v,t M ζ denote the horzontal and vertcal lfts to T ζ T M, ζ T x M, of a vector X T x M, respectvely. Proposton.1. The operator R has the followng propertes. 1) R s a lnear somorphsm of H LM) onto H T M. Moreover, R ξ X h = X h,t M. ) R s a lnear somorphsm of V onto V T M and R s dentcally eqal zero on V j for j. 3) There s a decomposton V LM) = V 1... V n. Proof. Easy comptatons left to the reader..1) By Proposton.1, we have natral dentfcatons H LM) H T M T M X h X h,t M X

3 ON THE GEOMETRY OF FRAME BUNDLES 199 and.) V V T M T M X v, X v,t M X Hence, we have defned the vertcal lft X v, V, LM), of the vector X T x M, π LM) ) = x, satsfyng the property R X v, = X v,t M. Let c = c 1,..., c n ) R n and C = c j ) be n n matrx. We assme that the n + 1) n + 1) matrx ) 1 c C = c C s symmetrc and postve defnte. Let g T M be a Remannan metrc on T M. Now, we are able to defne a new class of Remannan metrcs ḡ = ḡ C on LM). Let F : LM) T M be any smooth fncton. Pt ḡx h, Y h ) = g T M X h,t M, Y h,t M ) F ), ḡx h, Y v, ) = c g T M X h,t M, Y v,t M ) F ), ḡx v,, Y v,j ) = c j g T M X v,t M, Y v,t M ) F ). Fx LM). Let e 1,..., e n be a bass n T x M, π LM) ) = x, sch that e 1 ) h,t M F ),..., e 1) h,t M F ) s an orthonormal bass n HF T M ). Then.3) e h 1,..., e h n, e v,1 1,..., ev,1 n,..., e v,n 1,..., e v,n n s a bass n T LM). Let G be a matrx of the Remannan metrc g T M wth respect to the bass e h,t M 1,..., e h,t n M, e v,t M 1,..., e v,t n M. The fact that ḡ s postve defnte follows from the followng lemma. Lemma.. Let ) I g hv G = g vh ĝ be a postve defnte symmetrc n n block matrx. Then the matrx ) I c g Ḡ = vh c g hv C ĝ s postve defnte. Proof. It sffces to show that each prncpal mnor Ḡk, k = 1,..., n + n, of Ḡ s postve. Obvosly det Ḡk = 1 > 0 for k = 1,..., n. Hence we assme k > n. Then each mnor Ḡk s of the same form as the whole matrx Ḡ, ths we wll make calclatons sng matrx Ḡ. Comptng the determnant of the block matrx we get det Ḡ = det C ĝ c g vh )c g hv ) ) = det C ĝ c c) g vh g hv ) ) = det C c c) ĝ + c c) ĝ g vh g hv ) ).

4 00 K. NIEDZIAŁOMSKI Snce det C c c ) = det C > 0, det ĝ > 0, det c c ) 0, det ĝ g vh g hv) = det G > 0, t follows that matrces C c c) ĝ and c c) ĝ g vh g hv ) are postve defnte. Hence thers sm s postve defnte. If C = I and g T M s the Sasak metrc, then we get Sasak Mok metrc. Assme now C = I and g T M s a natral Remannan metrc on T M [8, 1] sch that g T M X h, Y h ) = gx, Y ) and dstrbtons H T M, V T M are orthogonal. Hence, there are two smooth real fnctons α, β : [0, ) R sch that ḡx h, Y h ) = gx, Y ),.) ḡx h, Y v, ) = 0, ḡx v,, Y v,j ) = 0, j, ḡx v,, Y v, ) = α F ) ) gx, Y ) + β F ) ) g X, F ) ) g Y, F ) ). The above Remannan metrc does not see the ndex of the dstrbton V. Snce all dstrbtons H LM), V 1,..., V n are orthogonal, t follows that we may pt F ) = n the last condton, that s consder a famly of maps F 1,..., F n rather than one map F. Then we obtan the postve defnte blnear form, hence the Remannan metrc, of the form ḡx h, Y h ) = gx, Y ),.5) ḡx h, Y v, ) = 0, ḡx v,, Y v,j ) = 0, j, ḡx v,, Y v, ) = α )gx, Y ) + β )gx, )gy, ). Now, we defne fnctons α, α : LM) R etc. as follows α ) = α ), α ) = α ), etc. To make the formlas n the next secton more concse, we wll wrte α, α etc. nstead of α ), α ) etc. 3. Geometry of ḡ Let M, g) be a Remannan manfold, LM), ḡ) ts frame bndle eqpped wth the metrc ḡ of the form.5). Let and R denote the Lev Cvta connecton and the crvatre tensor of ḡ, respectvely.

5 ON THE GEOMETRY OF FRAME BUNDLES 01 We recall the denttes concernng Le bracket of horzontal and vertcal vector felds [9] [X h, Y h ] = [X, Y ] h RX, Y ) ) v,, [X h, Y v, ] = X Y ) v,, 3.1) [X v,, Y v,j ] = 0. Moreover, n the local coordnates, for X = ξ x we have 3.) 3.3) X h j ) = a,b X v,k j ) = ξ jδ k Γ j ab a ξ b where Γ j ab are Chrstoffel s symbols [9]. Proposton 3.1. Connecton satsfes the followng relatons X hy h) = XY ) h 1 RX, Y ) ) v, X hy v,) = α R, Y )X) h + XY ) v, X v,y h) = α R, X)Y ) h X v,y v,j) = 0 j, X v,y v,) = α gx, )Y v, + gy, )X v, ) α where U = v,. + β α α β α α + β ) gx, )gy, ) + Proof. Follows from the formla for the Lev Cvta connecton ḡ A B, C) = AḡB, C) + BḡA, C) CḡA, B) relatons 3.1) and the followng eqaltes β α ) α + gx, Y ) U β, + ḡ[a, C], B) + ḡ[b, C], A) + ḡ[a, B], C) X v, g, Y )) = gx, Y ), X v, ) = gx, ), X h g, Y )) = g, X Y ). Before we compte the crvatre tensor, we wll need some formlas concernng the Lev Cvta connecton of certan vector felds.

6 0 K. NIEDZIAŁOMSKI Lemma 3.. The followng eqaltes hold X hu ) = 0, X v,u j) = 0 j, and X v,u ) = α + α X v, α + α β α β ) + α β α α + gx, )U. β ) W R, X)Y ) Q) = j W j )R, X)Y ) Q + j j W R x j, X)Y ) Q) for any W T LM), where Q denotes the horzontal or vertcal lft. Proof. Follows by standard comptatons n local coordnates. Proposton 3.3. The crvatre tensor R at LM) satsfes the followng relatons RX h, Y h )Z h = RX, Y )Z ) h 1 + Z R)X, Y ) ) v, 1 α R, RY, Z) )X R, RX, Z) )Y R, RX, Y ) )Z) h, RX h, Y h )Z v, = RX, Y )Z) v, + α X R), Z)Y Y R), Z)X ) h α ) v,j RX, R, Z)Y ) j RY, R, Z)X) j j + α gz, ) ) v, β α RX, Y ) α α + g ) RX, Y )Z, U, β RX h, Y v, )Z h = α X R), Y )Z ) h 1 ) v, RZ, X)Y + α gy, ) ) v, α RX, Z) α β α α + β ) g RX, Z)Y, ) U, j RX, R, Y )Z) j ) v,j

7 ON THE GEOMETRY OF FRAME BUNDLES 03 RX h, Y v, )Z v,j = α α j R, Y )R j, Z)X ) h RX h, Y v, )Z v, = α gz, )R, Y )X gy, )R, Z)X ) h α RX v,, Y v, )Z h = α RX, Y )Z ) h R, Y )R, Z)X ) h α ) h RY, Z)X + α R, X)R, Y )Z R, Y )R, X)Z ) h + α gx, )R, Y )Z) h gy, )R, X)Z) h) RX v,, Y v,j )Z h = α α j R, X)R j, Y )Z R j, Y )R, X)Z ) h RX v,, Y v, )Z v, ) = C gx, )gy, Z) gy, )gx, Z) ) U + A gy, )gz, ) + B gy, Z) ) X v, A gx, )gz, ) + B gx, Z) ) Y v, RX v,, Y v,j )Z v,k = 0 f {, j, k} > 1 where A = 3α ) α α α + α β α β )α + α ) α α +, β ) B = α β α α α ) α α +, β ) α C = α + β + 3α α ) + α ) β + α β α β + α α β α α + β ) ) Proof. Follows from the characterzaton of the Lev Cvta connecton and Lemma 3.. Remark 3.. Notce that whch s eqvalent to the condton A α Bβ = C α + β ), ḡ RX v,, Y v, )Z v,, W v, ) = ḡ RZ v,, W v, )X v,, Y v,). Corollary 3.5. Let X, Y be two orthonormal vectors n the tangent space T x M. Then the scalar crvatre K of LM), ḡ) at LM), π LM) ) = x, and K of

8 0 K. NIEDZIAŁOMSKI M, g) at x M are related as follows KX h, Y h ) = KX, Y ) 3 α RX, Y ), KX h, Y v, ) = α α + β gy, ) ) R, Y )X, KX v,, Y v, ) = A gx, ) + gy, ) ) + B α + β gx, ) + gy, ) ), KX v,, Y v,j ) = 0 j. Corollary 3.6. If M, g) s of constant sectonal crvatre κ, then KX h, Y h ) = κ 3 κ α gx, ) + gy, ) ), KX h, Y v, ) = κ α gx, ) α + β gy, )) 0. If, n addton, αt )t < 3κ for all t > 0, then KX h, Y h ) > 0. Proof of Corollary 3.5 and 3.6. The formla for K follows by Proposton 3.3. Assme now M, g) s of constant sectonal crvatre κ and αt )t < 3κ for all t > 0. Snce gx, ) + gy, ), then KX h, Y h ) κ 3 κ α > 0. Corollary 3.7. The scalar crvatre s of LM), ḡ) at LM) s of the form s = s 1 α k Re, e j ) k + n 1),j,k k k C k + nb k α k. where s s the scalar crvatre of M, g) at x M and e 1,..., e n s an orthonormal bass n T x M, π LM) ) = x. Proof. Fx LM) and let e 1,..., e n be an orthonormal bass n T x M, π LM) ) = x. Consder a bass of T LM) of the form.3). Pt ḡj k = ḡe v,k, e v,k j ) = α k δ j + β k ge, k )ge j, k ). The nverse matrx ḡ j k ) to ḡk j ) s the followng ḡ j k = 1 α k δ j β k α k α k + k β k ) ge, k )ge j, k ).

9 ON THE GEOMETRY OF FRAME BUNDLES 05 Hence s = ḡ Re h, e h j )e h j, e h ) +,j,j,l,k + ḡ p v,k ḡ Re, e v,k j,j,k,l,p k ḡjl k ḡ jl k ḡ Re h, e v,k j )e v,k l, e h ) )e v,k l, e v,k p ) The formla for s follows now by Proposton 3.3, Remark 3. and the eqalty Re, e j ) k = R k, e j )e ).,j,j In the end, we show that, n the case of a Cheeger Gromoll type metrc over the manfold of constant sectonal crvatre, the sectonal crvatre of LM) s nonnegatve. Corollary 3.8. Assme Then αt) = βt) = t, t > 0. KX v,, Y v, ) = gx, ) + gy, ) ) ) 1 + gx, ) + gy, ) ) In partclar, f M, g) s of constant sectonal crvatre 0 < κ < 3n, then the sectonal crvatre K s nonnegatve. Proof. We have αt )t = t 1 + t < 3κ for all t > 0 f and only f 0 < κ < 3n. Hence, by Corollary 3.6, KX h, Y h ) 0 for X, Y T x M nt and orthogonal. Moreover, gx, ) + Y, ). Ths KX v,, Y v, ) ) = 1 + ) > 0. Acknowledgement. The athor wold lke to thank Professor Oldřch Kowalsk for valable sggestons.. References [1] Abbass, M. T. K., Sarh, M., On natral metrcs on tangent bndles of Remannan manfolds, Arch. Math. Brno) 1 1) 005), [] Bard, P., Wood, J. C., Harmonc morphsms between Remannan manfolds, Oxford Unversty Press, Oxford, 003. [3] Benyones, M., Lobea, E., Wood, C. M., Harmonc sectons of Remannan vector bndles, and metrcs of Cheeger Gromoll type, Dfferental Geom. Appl. 5 3) 007), [] Benyones, M., Lobea, E., Wood, C. M., The geometry of generalsed Cheeger-Gromoll metrcs, Tokyo J. Math. 3 ) 009),

10 06 K. NIEDZIAŁOMSKI [5] Cordero, L. A., Leon, M. de, Lfts of tensor felds to the frame bndle, Rend. Crc. Mat. Palermo ) 3 ) 1983), [6] Cordero, L. A., Leon, M. de, On the crvatre of the ndced Remannan metrc on the frame bndle of a Remannan manfold, J. Math. Pres Appl. 9) 65 1) 1986), [7] Dombrowsk, P., On the geometry of the tangent bndle, J. Rene Angew. Math ), [8] Kowalsk, O., Sekzawa, M., Natral transformatons of Remannan metrcs on manfolds to metrcs on tangent bndles. A classfcaton, Bll. Tokyo Gakge Unv. ) ), 1 9. [9] Kowalsk, O., Sekzawa, M., On crvatres of lnear frame bndles wth natrally lfted metrcs, Rend. Sem. Mat. Unv. Poltec. Torno 63 3) 005), [10] Kowalsk, O., Sekzawa, M., On the geometry of orthonormal frame bndles, Math. Nachr ), no. 1, ) 008), [11] Kowalsk, O., Sekzawa, M., On the geometry of orthonormal frame bndles. II, Ann. Global Anal. Geom. 33 ) 008), [1] Mok, K. P., On the dfferental geometry of frame bndles of Remannan manfolds, J. Rene Angew. Math ), Department of Mathematcs and Compter Scence, Unversty of Łódź, l. Banacha, Łódź, Poland E-mal: kamln@math.n.lodz.pl