On a Laplacian which acts on symmetric tensors

Transcription

1 On a Lalacan whch acts on metrc tensors keš J. Deartment of Algebra and Geometry, Palacky Unversty, 7746 Olomouc, Lstoadu, Czech Reublc e-mal: osef.mkes@uol.cz Steanov S. E., Tsyganok I.I. Deartment of athematcs, Fnance Unversty, 5468 oscow, Lenngradsky Prosect, 49-55, Russan Federaton e-mal: s.e.steanov@mal.ru,..tsyganok@mal.ru Abstract. In the resent aer we show roertes of a lttle-known Lalacan oerator actng on metrc tensors. Ths oerator s an analogue of the well known Hodge-de Rham Lalacan whch acts on exteror dfferental forms. oreover, ths oerator admts the Wetzenböck decomoston and we study t usng the analytcal method, due to Bochner, of rovng vanshng theorems for the null sace of a Lalace oerator admttng a Wetzenböck decomoston and further of estmatng ts lowest egenvalue. Key words: Remannan manfold, second order elltc dfferental oerator on metrc tensors, egenvalues and egentensors. SC00: 53C0; 53C; 53C4. Introducton In resent aer we show roertes of a lttle-known dfferental oerator of second order whch acts on metrc tensors on a Remannan manfold (, g). Ths oerator s an analogue of the well known Hodge-de Rham Lalacan whch acts on exteror dfferental forms (see [,. 54]; [,. 04]). The oerator s a self-adont Lalacan oerator and ts kernel s a fnte-dmensonal vec- tor sace on a comact Remannan manfold (, g). In addton, the Lalacan admts the Wet- zenböck decomoston and so we can study t usng the analytcal method, due to Bochner, of rovng vanshng theorems for the null sace of a Lalace oerator admttng a Wetzenböck decomoston and further of estmatng ts lowest egenvalue (see [,. 53]; [,. ]; [3]; [4] and [5]). The aer s organzed as follows. In the next secton, we gve a bref revew of the Remannan geometry of the Lalacan. In the thrd secton of the aer we rove vanshng theorems of the ker-

2 nel of. In the fourth secton, we demonstrate the man result of the resent aer whch s the es- tmaton of the frst egenvalue of the Lalacan for the Yano Lalacan wll be obtaned.. In the fnal secton a Bochner ntegral formula. Defntons and notatons.. Let (, g) be a comact orented C -Remannan manfold of dmenson n and S metrc tensor roduct of order of a tangent bundle T of. On tensor saces on we have be a canoncal scalar roduct g(, ) and on ther C -sectons the global scalar roduct,. In artcu- lar, for any, С S we have, g, dv, (.)! where dv s the volume element of (, g). If g,... and... denote comonents of the metrc g and tensor felds, С S wth resect to a local coordnate system n x,...,x on (, g) then!...,... dv... where g...g and... g k g k for the Kronecker delta. Next, f D s a dfferental oerator between some tensor bundles over, ts formal adont D * s unquely defned by the formula D,,D (see [,. 460]). For examle (see [,. 54]), the covarant dervatve C S C T S by : C T S C S... g... :, whose formal adont for an arbtrary C T S wll be also denoted. In local coordnates, ths oerator s defned by equaltes. Furthermore we defne (see also [,. 54]) the oerator : C S C S of degree by the formula Sym for an arbtrary С S. In local coordnates we have

3 Then : C S C S s the adont oerator of : wth resect to the global roduct (.)... In [6] the oerator С S С S of degree was defned. The oerator s related to a varatonal roblem (see [6]), as follows: If we defne the energy of metrc tensor feld by E ½,, then 0 s the condton for a free extremal of E ½,. Ths oerator s an analogue of the well known Hodge-de Rham Lalacan : С М С М whch acts on C -sectons of the bundle of covarant skew-metrc tensors of degree ; n other words, exteror dfferental -forms on (see [,. 34]; [,. 04]). The oerator s a self-adont Lalacan oerator and ts kernel s a fnte-dmensonal vector sace on a comact Remannan manfold (, g) (see [6]; [7], [0]). In ths case, by vrtue of the Fredholm alternatve (see [,. 464]; [,. 05]) vector saces Ker and Im are orthogonal com- lements of each other wth resect to the global scalar roduct (.),.e. C S Ker Im. (.) At the same tme, we recall that ths decomoston s an analog of the well-known Hodge orthogonal decomoston C Ker Im where п and the kernel of conssts of harmonc exteror dfferental -forms on (see [],. 0-3). Comare the oerator wth the rough Lalacan (see [,. 54]). Frst, t s easy to see that these two oerators concde f (, g) s a locally Eucldean sace. Second, the oerator has order zero and can be defned by metrc endomorhsm B of the bundle S where B can be algebracally (even lnearly) exressed through the curvature and Rcc tensors of (, g) (see [,. 53]; [6]). Ths can be exressed by the Wetzenböck decomoston formula means that s a Lchnerowcz Lalacan. In local coordnates we have s... g r... k k s k... k k l s mt m k l... k s k... l t l... B and B g g R, (.3)

4 where r and R kl are comonents of the Rcc tensor Rc and curvature tensor R wth resect to a local coordnate system n x,...,x. Remark. Egenvalues wth ther multlctes of the Lchnerowcz Lalacan L + B, whch acts on the vector sace С S over the Eucldan shere n S were dscussed n []. In artcular, f (, g) s a locally Eucldean manfold then the equaton 0 becomes k x... k 0 wth resect to a local Cartesan coordnate system n x,...,x. Ths means that all comonents of a covarant comletely metrc -tensor on a locally Eucldean manfold (, g) are harmonc functons belongng to the kernel of the Lalacan. Ths roerty s also characterstc for any harmonc exteror dfferental -form (see [,. 05-]). Therefore, the metrc tensor ker was named n [6] as a harmonc metrc -tensor on a Remannan manfold (, g). Remark. For a harmonc metrc -tensor, t was sad n [6] that t does not have any geometrcal nterretaton for. It was also onted out that metrc tensors, whose covarant dervatve vanshes s clearly harmonc, the metrc tensor g beng the most mortant examle. These tensors are trval examles of a harmonc metrc tensor. Now we can say (see Theorem 4.6 n the aer [] of the frst of the three authors of ths aer) that the 3-tensor T of a nearly ntegrable SO(3)-structure on a fve-dmensonal connected Remannan manfold (, g) s a non-trval examle of a harmonc metrc tensor..3. For, the exlct exresson for s suffcently comlcated but for the case, t has the form B where B Rc. From well known Wetzenböck decomoston formula Rc we obtan Rc, where, n local coordnates, we have ( r (see [7]; [0]). Ths form of the oerator was used by K. Yano (see, for examle, the monograh [8,. 40] and aer [9]) for the nvestgaton of nfntesmal sometres of

5 (, g) reservng the metrc g and secal vector felds named geodesc vector felds. Therefore, n aer [0], we called t as the Yano oerator or, n other words, the Yano Lalacan. We wll use ths name for throughout ths aer. On the other hand, a vector feld X on (, g) s called an nfntesmal harmonc transformaton f the one-arameter grou of local transformatons of (, g) generated by X conssts of local harmonc dffeomorhsms (see [3]). At the same tme, the equalty 0 for gx, s necessary and suffcent condton for a vector feld X to be an nfntesmal harmonc transformaton on (, g) (see [7], [0]). Hence, n the secal case of the kernel of conssts of nfntesmal harmonc transformatons. In addton, we recall the followng examles of nfntesmal harmonc transformatons: nfntesmal sometrc transformatons of Remannan manfolds (see [8,. 44]), nfntesmal conformal transformatons of two-dmensonal Remannan manfolds (see [8,. 47]), affne Kllng (see [8,. 45]) and geodesc vector felds (see [9]) on Remannan manfolds, holomorhc vector felds on nearly Kähleran manfolds (see [0]) and vector felds that transform a Remannan metrcs nto Rcc solton metrcs (see [7]). In concluson, we ont out that the vector sace Ker : Ker Ker of all nfntesmal sometrc transformatons must be an orthogonal comlement of the vector sace Ker Im d of all nfntesmal gradent harmonc transformatons wth resect to the whole sace Ker on a comact orentable Ensten manfold (, g) (see [9]). In vew of (.) we can wrte the followng orthogonal decomoston C S Ker Im d Ker * Im. 3. Vanshng theorems 3.. Let (, g) be a comact and orented Remannan manfold of dmenson n and : С S С S be the Yano Lalacan. Then the followng rooston s true (for the roof see Aendx).

6 Lemma 3.. Let (, g) be a comact orented C -Remannan manfold of dmenson п and : С S С S be the Yano Lalacan then,,, В (3.) where В, g B!, dv. The ntegral formula (3.) we obtan by the Wetzenböck decomoston formula artcular, (.3) becomes B In В,!... r... R kl k... 3 l... 3 dv wth resect to a local coordnate system n x,...,x. For the secal case of we can rewrte the formula (3.) n the form,, Rc X, X (3.) where gx, for an arbtrary X C T *. It s well known that Rc. Then from (3.) we obtan the formula,, Rc X, X. (3.3) Corollary 3.. Let (, g) be a comact orented C -Remannan manfold of dmenson п and : С S С S be the Yano Lalacan then,, Rc X, X where g X, for an arbtrary X C T *. 3.. The curvature tensor R defnes a lnear endomorhsm R : S S by the formula R l R kl k where k l Rkl are local comonents of R and g g kl for local comonents of an arbtrary C S. The metres of R mly that R s a self-adont oerator, wth resect to

7 the ontwse nner roduct on S. Hence the egenvalues of R are all real numbers at each ont х М. Thus, we say R s ostve (res. negatve), or smly R > 0 (res. R < 0), f all the egenvalues of R are ostve (res. negatve). Ths endomorhsm R : S S was named as the oerator of curvature of the second knd (see [5]). Remark. Defnton, roertes and alcatons of R can be found n the followng monograh and aers []; [4]; [5]; [6]; [7] and etc. Let (, g) be an n-dmensonal Remannan manfold. If there exsts a ostve constant such that Rkl l k (3.4) holds for any C S, then (, g) s sad to be a Remannan manfold of negatve restrcted curvature oerator of the second knd. In ths case the followng theorem s true. Theorem 3.3. Let (, g) be an n-dmensonal comact orented Remannan manfold. Suose the curvature oerator of the second knd R : S S s negatve and bounded above by some negatve number at each ont -tensors. х М. Then (, g) does not admt non-zero metrc harmonc Proof. Let be a harmonc metrc -tensor then from (3.) we obtan В,, 0. (3.5) On the other hand, from (3.4) we obtan the followng nequaltes whch show that k3... l k l R R... kl 3... lk... n r......,, п, 3... В < 0 (3.6) ;

8 for an arbtrary non-zero C S. Inequaltes (3.5) and (3.6) contradct each other. Ths contradcton roves our theorem. Remark. We recall that an orentable and comact Remannan manfold (, g) does not admt a non- zero harmonc exteror dfferental -form f ts curvature oerator R : S S s ostve and bounded below (see [5]) A -dmensonal mmersed submanfold of (, g) s called a geodesc f exsts a arameterzaton : x t x for t I R satsfyng x 0. If each soluton x x t x of the equatons x x 0 of the geodescs satsfes the condton x,...,x const for smooth covarant comletely metrc -tensor and k d x x, then the equatons k x x 0 d t х admt so called a frst ntegral of the -th order of dfferental equatons of geodescs. The equaton X,X,...,X Х,Х,...,Х 0 Х serves as necessary and suffcent condton for ths [8,. 8-9]. On the other hand, the ntegral formula (3.) can be gven n the form,,,, 0 В. (3.7) Thus, by the Theorem 3., we have the followng result (see also [9]). Corollary 3.4. Let (, g) be an n-dmensonal comact orented Remannan manfold. Suose the curvature oerator of the second knd R : S S s negatve and bounded above by some negatve number at each ont of dfferental equatons of geodescs. х М. Then (, g) does not admt frst ntegrals of the -th order Fnally, for the secal case of, by the Corollary 3., we have (see also [0]) Corollary 3.5. If, n n-dmenson п comact orentable Remannan manfold (, g), the Rcc tensor Rc s negatve defnte, then (, g) does not admt non-zero nfntesmal harmonc transformatons.

9 4. Sectral roertes of the Yano Lalacan 4.. A real number r, for whch there s a tensor С S (not dentcally zero) such that, s called an egenvalue of and the corresondng С S an egentensor of corresondng to. The zero -tensor and the egentensors corresondng to a fxed egenvalue form a vector subsace of S denoted by V and called the egensace corresondng to the egenvalue. The followng theorem s true. Theorem 4.. Let (, g) be an n-dmensonal п comact and orented Remannan manfold and : С S С S be the Yano Lalacan. ) Suose the curvature oerator of the second knd R : S S s negatve and bounded above by some negatve number at each ont х М. Then an arbtrary egenvalue of s ostve. ) The egensaces of are fnte dmensonal. 3) The egenforms corresondng to dstnct egenvalues are orthogonal. Proof. ) Let V be a non-zero egentensor corresondng to the egenvalue, that s then we can rewrte the formula (3.) n the form, В,,. (4.) If we suose that the curvature oerator of the second knd R : S S s negatve and bounded above by some negatve number at each ont х М then, by the nequalty (3.), we conclude, В,, п,, 0. (4.) ) The egensaces of are fnte dmensonal because s an elltc oerator.

10 3) Let and, be the corresondng metrc egentensors. Then,, and,,,. Therefore 0, and snce t follows that, 0, that s, and are orthogonal. Usng the general theory of elltc oerators on comact (, g) t can be roved that has a dscrete sectrum, denoted by Sec () (), consstng of real egenvalues of fnte multlcty whch accumulate only at nfnty. In bols, we have Sec () () In addton, f we suose that the curvature oerator of the second knd R : S S s negatve and bounded above then Sec () () oreover, the followng theorem s true. Theorem 4.. Let (, g) be an n-dmensonal comact and orented Remannan manfold. Suose the curvature oerator of the second knd R : S S s negatve and bounded above by some negatve number at each ont х М. Then the frst egenvalue of the Yano Lalacan : С S С S satsfes the nequalty n. Proof. Suose the curvature oerator of the second knd R : S S s negatve and bounded above by some negatve number at each ont. Then for an egentensor corresondng to an egenvalue λ, (4.) becomes the nequaltes х М, п,, п, (4.3) whch rove that n 0 If the equalty s vald n (4.4), then from (4.3) we obtan 0 the dentty 0 В s strctly negatve for an arbtrary nonzero С S. Ths comletes the roof., because. (4.4) and В 0. In ths case we have

11 In the secal case of we obtan the followng corollary from the above theorem. Corollary 4.3. Let (, g) be an n-dmensonal comact orented Remannan manfold. Suose the Rcc tensor satsfes the nequalty Rc n g for some ostve number, then the frst egenvalue of the Yano Lalacan : С S С S satsfes the nequalty п. The equalty п s attaned for some harmonc egenform C S and n ths case the mul- tlcty of s less than or equals to the Bett number b (). Proof. Let (, g) be an n-dmensonal comact and orented Remannan manfold, and defne the Rcc tensor Rc of (, g) by ts comonents r wth resect to a local coordnate system there exsts a ostve constant such that n x,...,x. If holds for any curvature. C r n (4.5) S, then (, g) s sad to be a Remannan manfold of negatve restrcted Rcc Next, suose that the Rcc s negatve and bounded above by some negatve number п at each ont х М,.e. Rc n g, then from Rc we obtan the nequalty for an arbtrary, (4.6) becomes the nequaltes C, п,, (4.6) S. Then for an egenform corresondng to an egenvalue, п,, п, (4.7) whch rove that 0 n. (4.8) If the equalty s vald n (4.8), then from (4.7) we obtan 0. In ths case s a harmonc -form and the multlcty of s less than or equals to the Bett number b (). 4. Aendx

12 In ths secton we rove the Lemma 3.. For ths we ntroduce a vector feld X wth comonents wth resect to a local coordnate system n x,...,x. Then dv X k3... R l r... kl In turn, for the vector feld Y wth local comonents, we have the equaltes... dv Y. Let (, g) be a comact and orented Remannan manfold then t follows from the Green s theorem dv Z dv 0 for Z X Y that... 3 r... R 3 kl k... l dv 0... (4.) Snce the ntegral formula (4.) can be wrtten n the form where В,,,,, 0 В (4.) g B!, dv! ! r R ,... 0 kl k... 3 dv ; l... 3 dv ;

13 ......,... dv ;, dv....! 0 0! М The oerator satsfes the roerty,,,, whch follows mmedately from ts defnton. Therefore, the ntegral formula (4.) can be rewrtten n the form (3.). References [] Besse A.L, Ensten manfolds, Srnger-Verlag, Berln Hedelberg (987). [] Petersen P., Remannan Geometry, Srnger Scence, New York (006). [3] Wu H., The Bochner technque n dfferental geometry, New York, Harwood Academc Publshers (988). [4] Chavel I., Egenvalues n Remannan Geometry, Academc Press. INC, Orlando (984). [5] Craoveanu., Puta., Rassas T.., Old and new asects n sectral geometry, Kluwer Academc Publshers, London (00). [6] Samson J.H., On a theorem of Chern, Transactons of the Amercan athematcal Socety 77 (973), [7] keš J., Steanov S.E., Tsyganok I.I., From nfntesmal harmonc transformatons to Rcc soltons, athematca Bohemca 38: (03), [8] Yano K., Integral formulas n Remannan geometry, arcel Dekker, New York (970). [9] Yano K., Nagano T., On geodesc vector felds n a comact orentable Remannan sace, Comment. ath. Helv. 35 (96), [0] Steanov S.E., Shandra I.G., Geometry of nfntesmal harmonc transformatons, Annals of Global Analyss and Geometry 4 (003), [] Boucetta., Sectra and metrc egentensors of the Lchnerowcz Lalacan on S n, Osaka J. ath., 46 (009), [] keš J., Steanova E.S., A fve-dmensonal Remannan manfold wth an rreducble SO(3)- structure as a model of abstract statstcal manfold, Annals of Global Analyss and Geometry, 45: (04), -8.

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