ON CONFORMAL KILLING TENSORS OF DEGREE 2
． り IN KAHLERIAN SPACES BYMAS▲MI SEKIZAWA
Dedicated to Prof． Dr． Tyuzi Adati on the occasion of his sixtieth birthdayIntroduction． Let・Mbe an n・dimensional Riema㎜ian space with the
metric tellsor gdi． A vector field vi＊）is called a Killillg vector if it satis丘es 7〆Vi十ク‘vゴ＝0， where Vi means the operator of the covariant differentiation with respect to the Rieman垣an comection． We denote by Rkii厄the Riemannian curvature tensor and by Rli＝RaJia， RニθbaRba the Ricci tensor， the scalar curvature respectively． AKilling tensor吻‘of degree 2 is a skew−symmetric tensor丘eld satisfying l7kUdi十7」2tki＝0． S．Tachibana［6］has de丘ned a conformal Killing tensor of degree 2 as a skew・symmetric tensor丘eld％ゴ‘satisfying （1） 7鳶μが十7」％永‘＝2ρtgkゴーρk9ゴi一ρdgk‘， whereρ‘is a certa血vector丘eld． Such a vector丘eldρ‘is’ モ≠撃撃?п@an as・ sociated vector of％，‘and evidently it can be written as （2） （n−1）ρt＝7aμat． Especially， ifρ‘vanishes identically， a conformal Killing tensor吻is re・ duced to a Kユlhng one． For a conformal Killmg tensor of degree 2， the following identity is known （Tachibana［6］p．59） （3） （n−2）（7ノρ‘十7‘ρゴ）＝R∫aUta十R‘α％ゴα． The purpose of the present paper is to discuss the conformal Kinmg tensor of degree 2 in a Kahlerian space． In§2we shall show that the pure collformal Killing tensor of degree 2 is Ki1血1g one．§3will be devoted to the discussion of the hybrid conformal Killing tensor of degree 2． In§4 we shall give an example of the conformal Killing tensor of degree 2 in a Kahlerian space with parallel Boc㎞er curvature tensor and with a non constant scalar curvature． 1．Klihlerian space． A K員hlerian space is an even dimensional space with ＊）We adopt the identi丘cation of a vector且eld with a 1−form by virture of the Riemannian metric．［1］
2．．、． ・．．』』 s・．1．．．． M．SEKIZAWA ． ． ’ ・「 ニ ゴ やごサレシドダロ amixed tensor pih and with a Riemamlian metric tensor』σゴi satisfying conditions： 9t“9・h・ ：一δih，9b。9ゴbgiaニσゴi anα7ゴ伜九＝o． If we put伽＝Pゴα9砺then we can see that ∼ρゴε＝一乎）‘フ and ク碑）ガ＝0． It is well known that there hold the followi皿g relations： Rkグiag・h・＝Rkj・hgia， R。xngkaニーR此。紳ゴα， （4） Riag・n＝R・hgia， Rj。9ia＝−R。砺α．． For the tensor Sゴi defined by Sゴi＝gゴα．Rai， we have the foUowmg relations： S5声一S‘，， 27αSゴα＝9ia7。R， （5） ． Sゴ‘＝：一（1／2）乎）baRゴiba＝gbaRbliq， sゴ・ρia＝−Saig」a＝Rji， gbaSba＝−R． Atensor幻が（resp． viりis called to be pure， if it satis丘es the relation （6）． Vaig豆α＝±vゴa乎）ia（resp． Vahψia＝t）iagah） and to be hybrid， if it satisfies the relation （．7り η⇒ゴ・＝一σ∫。P、a（resp．砂。培・＝−Vi・9。・）． Avector血eld vn is called．a contravariant analytic vector， if its convariant derivative 7ivh is pUre， i．e． it satis丘es giav。vn＝．Paπ7ive． For the contravariant…malytic vector， the following Lemma is well㎞own ［8］： 1・EMMA． If a compact Kd九letiian spαce九α8 a negαt初e semi．definite Rεcci curvα彦％ゲθ， thenαeontravαriant analytic vector九α8 vanis九飢9 COVα吻nt derivative． 2．、Pure conformal Killing tensor of degree 2． Let us consider a conforma1 耳i1地g．tensor％ゴi of degree 2 which is．pUre．in．．anπ三dimensional Kahlerian space． TranseVecting（1）with gゴi and making use of（2）andψゴiμゴi＝O， we get 伽＋2）9kap。＝0， from whichρ鳶＝0． Thus we have THEgREM 1・・4 pure con∫ormal・Killing tensor o∫．（legree 2飢αKdhleriαn spαCθi8αKilling Oπθ． ’ 3・ Hybrid conformal Killing tensor of deg1℃e 2・ L6t us cσnsi｛享er a con’ formal Killing tensor吻るof degree 2 which is hybrid in an．n・dimellsional K註hlerian spaee M． Transvectillg（1）with gJi．and．making use of（2）and（7）， we get （8） 一．’． ’7kα＝一（n−4）9、・ρ。｝ where we putα＝＝ ipjiuji． Transvedting this equation with gik， We have
ON CONFORMAL、 KILLING T・ENSORS OF DEGREE 2 3 （9） 乎）ia7aα＝（n−4）ρz． Substituting the above equation to（3）， we obtain n−2 （ψzα7ゴ7αα十ψゴ47乞7αα）＝RゴaUi。＋RiαUf。， n−4 provided％≠4． Transvectmg this equation with侮z and using（4）and（7），
we have
慧一・…α＋卿〃…α）一一…Ra・ze・・b＋輌・・、・． Taking the symmetric part of the above equatiol1， we get n−2 （7iC7jcr一ψw〃δ’。α）＝0． n−4Hence we have
（10） 9、・7。クhα一9・，。九7、7・αニ0， provided n≧6．「rhen we have THEOREM 2・1カαn n（≧6）−d勿nensionαl Kdihlθriαn spαcθ， i∫ωθP砿α＝ρブ扱ゴ信 ∫0グαhybrid conformα1 Killing tensor彫‘o∫degree 2， thenク抱αisαcoπ． travαriant仇吻tic vector． C・R侃LARY・肋・n n（≧6）・dim・nsi・励κ醐θ・励・Pαcθ，助九θ九抑磁c・π・ formal Killing彦ensor zeji O∫〔》θσγθθ2is notαKilling ten80γ，九θπクゐαis α’ non−t物iαl con彦rαvari（城αn吻tic vector． Now， transvecting（10）withσゴπand usmg（9）， we get （n−4）（7」ρ兎十7是ρゴ）＝0．Thus we have
THEOREM 3． 1物an n（；≧6）−dimensionαl Kdihlθriαn 8pαcθ君んθαssociαte〔1 vector o∫hybrid conformαI Killing tensor o∫degree 2 isα Kil吻σvector． Let us assume that the n−dimensional Kahlerian space 1匠is compact and has a negative semi−definite Ricci curvature． Then， by L，emma stated m§1and Theorem 2クwe have
7i7hα＝0；from which
7α7￠α＝0． Since M is compact， using the Green，s Theorem， we can see thatαis a constallt in 1匠． Thus， from（9），ρる＝O provided n≠4． ． THEOREM 4．万an n（≠4）−dimensionαl co脚αct Kdhleri仇8Pαcθん〔tS・a・nega・ 励θSθ？ni−definite Ricci curvαture，孟九enαhy励d conformα1 Killing tenSOア0∫ degree 2 isα．KづZ吻g oπθ． 4．An example of the conformal Killing tensor of degree 2． In an n・ dimensional Kahlerian space the Bochner curvαture tensor is defined by4 M； SEKIZAWA． Tachibana〔4］as follows： KiCプih＝Rkjih十（1／（カ十4））（Rkiδゴlt−Rガδ北九十9iCiRゴ九一σゴiRkh 十S腕乎♂−Sdzg）kh十9kiSゴh−9ゴt・Skh十2S砺ρ♂十2伜，Siり 一（R／（n＋2）（n＋4））（σ碗δ’為一9ゴiδ・’b＋伽φ，庖一￠ゴi9、’b＋29、」P、n）． The ab・ve B・・㎞er curvature tens・・is the real representati・n・f the tensor introduced by Bochner［1］． An example of a Kahlerian． space with non・constant scalar curvature has been given in the space of vanishing B・chner curvature tens・r With certaih K置hler迦metric by Tachibana and Liu［7］． Therefore， in the rest of this section， we shall assume that the scalar curvature R is not a constant． Let．us consider an n．dimensional Kahlerian space M with the para1161 Bochmer’curvatUre tensor． The foUow． ing equation holds good（Matsumoto［2］． p．26） 7・R炉（1／2伽＋2））（σ・d7・R＋b・・7」R−9・jgi・7。R−P、、9〃。R＋20〃i、R）． From this and（5）we have 7，sゴ乞十7〆Ski＝−9ia（7kRゴa十7iRka） ＝（−1／2伽＋2））（2σ・・ep・a7・tR一卿・・7。R一σ、・卿。R） 十（3／2伽十2））（ψki7ゴR十ρゴる7盈R）
