ANALYTIC TENSOR AND ITS GENERALIZATION

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1 ANALYTIC TENSOR AND ITS GENERALIZATION SHUN-ICHI TACHIBANA (Received September 20, 1959) In our previous papers [7], [8], υ te notion of almost-analytic vector was introduced in certain almost-hermitian spaces. In tis paper we sall deal wit tensors and obtain te notion of Φ-tensors wic contains, as special cases, te one of analytic tensors and decomposable tensors. 1. Let us consider an w-dimensional space 2) wic admits a tensor field ψi 5 of type (1,1). Let (i) ϋ) = ; *! I «"" Λ be a tensor of type (q,p). If it commutes wit φ^, ten we sall say tat f (ί) ϋ) is pure wit respect to te corrresponding indices, namely it is pure wit respect to i k and 9 if (1) ξ ίp -r-h U) φ,ζ = W«- r "VΛ and pure wit respect to i k and i 9 if If Ico U) anti-commutes wit φ ten we sall say tat it is ybrid wit respect to te corresponding indices. Tus if for example, olds good, teα it is ybrid wit respect to ί k and. ξ^ϋ) is called pure (resp. ybrid) if it is pure (resp. ybrid) wit respect to all its indices. φ itself and S t are examples of te pure tensor. If ψi is a regular tensor i.e. det^/) =t= 0, ten te tensor wose components are given by te elements of te inverse matrix of iψi) is also pure. LEMMA I. If (ί) U) is pure {ybrid) wit respect to some indices, ten so υ) We sall prove only te case wen ξ {ί) (Jz 4= 1). In fact, we ave is pure wit respect to i Ύ and 1) Te number in brackets refers to te bibliograpy at te end of te paper. 2) We sall mean by a space a differentiable manifold of class C, and denote by its local coordinates. Indices run over 1, 2, n.

2 ANALYTIC TENSOR AND ITS GENERALIZATION 209 = lv / 2 ί ϋ) ψii q. e. d. LEMMA 2. If a skew-symmetric tensor? (ί ) is pure, ten (l) = p *-'Ur<P r is also a skew-symmetric pure tensor. * In fact, C ί) is pure by virtue of Lemma 1. It is evident tat it is skewsymmetric wit respect to i k and i (k, =f= 1). For k =+= 1, we ave * = ( I)*" 1 ς ίp...t k... ίv ψ ίk r= ( ) " 1 ^ * = - lv.. ir.. /2V q. e. d. tu) If ξ it) < = fd/v -Ί is a pure tensor of type (g + l,/>), ten u {i) also pure for a (covariant) vector z/s. Generalizing tis fact, we ave easily LEMMA 3. Let ω ω and η ( a) Φ) be pure tensors of type (q,p + 1) and (q + l 9 p) respectively. Ten {i) U) Vta) Φ) is also a pure tensor of type (q + q, p + p), provided tat pλ-p =4= 0 or q + q 4= 0. A tensor φ/ is called an almost-product structure, if it satisfies φΐφ δi J, and is called an almost-complex structure, if it satisfies φ t r φ = S^, [U [2], [4], [12]. In tese cases, we can verify te following lemmas. LEMMA 4. Let φ/ be a tensor suc tat φf φ r 5 ~ δi. 3) Ten we ave ζ r r 0 for a ybrid tensor ξ t \ LEMMA 5. Let φ/ be a tensor suc tat φΐ <p r = 8 t. If ξt (ξ i ) is pure and v/ivu) is ybrid, ten we ave is LEMMA 6. Let <p t be a regular tensor, i.e. rank (φ) = n. If ξ ki (ζ k /) is ybrid, ten it is a zero tensor. In fact, we ave ri ψΐ = ri ψk = %Tcr ψi = ~ Stsrt Ψi\ from wic we find ξ ki = 0. q. e. d. Now consider an almost-complex structure φ^, ten if we coose a suitable frame at a point, ψι as te following components at te point. 3) In tis paper, by ε we sall always mean ± 1. 4) Indices α, β, run over 1,..., m(= n/2) and α = m + α.

3 210 S. TACHIBANA to te equ- Wit respect to tis frame, te equation (1) is equivalent ations ; V* PΛ. * Λ= 0 - V Pλ Λ = 0»' p**' α fc'"*l u > bι p «Λ ii ' v, and te equation (2) is equivalent to fe,,. «* «.v ' >v Λ = o, f v.. ϊt..., I v? *- Λ = o. In tis sense we ave used te terminologies "pure" and "ybrid" [6], [11]. 2. An almost-hermitian space admits, by definition, a Riemannian metric tensor g t and an almost-complex structure φ suc tat g rs <p[<pi = da- A Kalerian space is an almost-hermitian one suc tat te equation (3) V&ΐ = 0 is valid, wsre v* denotes ts operator of te covariant derivative wit respect to te CristoffeΓs symbol..[. We sall devote tis section to a Kalerian space. ω A pure tensor ζ (ί) is called analytic [ 6 ], if its covariant yιζ(i) ϋ) is also pure, i.e. it satisfies (4) W - Vrf(<> ϋ) = 9>, I 'V,fv * ( ' ), or *» (P Vr </> <Λ = 0> derivative In fact, te equation (4) is equivalent to te following one wit respect to complex coordinates (z*, z*): Te definition (4) of te analytic tensor contains te Kalerian metric in appearence, but (5) is independent to te metric. Hence it is natural to ask if te notion of te analytic tensor is defined in a complex manifold wit respect to real coordinates. In tis point of view, we sall attempt to eliminate te CristoffeΓs symbols in (4) by making use of (3). If we write down (4) explicitly, we vae (6) = < [aa V +Σ U)

4 ANALYTIC TENSOR AND ITS GENERALIZATION 211 were we ave put d r = d/dx r. On te oter and, on taking account of (3), we ave If we substitute tese equations into (β) and take account of te purity of f(o ω, ten we find Σ ^? ' r - Λ = o, were f (/) () is defined by ί 8 ) (i) <» = φ tl r ξ ip,,/«= φ* ξ ω} «K 3. As we ave known in te preceding section, te equation (7) wic defines te analytic tensor in a Kalerian space does not contain te Kalerian metric. Following to tis fact, we sall introduce an operator in a space wic admits a tensor field of typs (l, 1). Te operator will produce from a pure tensor of type (q, p) a new tensor of type (q, p +1). Let ψι be a tensor of type (1, 1) and (o (i) a pure tensor of type (#,/>). Now we define an operator Φ by V" * were f(/) (/) is given by (8). In te rest of te present section, we sall sow tat Φ^,/ 0 is a tensor, U) if ξ V) is pure. Let T)ι be an affine connection, S t its torsion tensor, i.e. S/ 1 = (1/2) (Tβ Γ?;) and by v& w e sall denote te operator of covariant derivative wit respect to Γ%. Hence if v is a vector field, ten its covariant derivative is given by y k v = 3*. v + IY r v". (i) If we represent (9) by terms of covariant derivatives, Φ, (0 is te sum of te following five terms a 19, a δ :

5 212 s TACHIBANA «i = φtzλ,» Q) = φi [v t f(«) ϋ) - 2IV* f w «2 =- 3 t ld) (Λ = - V«?(0 (Λ + SΓ^IioV ' *. = - Or ψ ξω'"- r - = 2 [- Vr ψΐ* + Γ ίf φl - Γ n?»/*] <,'«r ; Λ. If we denote te λ-t term of ώ^ by a^, te following relations old. Tus we find tat fc = l Q wic sows tat Φ t (ί) (/) is a tensor. 4. In tis section, we sall represent (9) in different forms. Using te notation in 3, we ave (12) α 4 = q 3, f cl) > - Σ ^,> 3 f (l /«."-- Λ Now if we put ten, substituting (11) into (9), we find tat (13) Φ ; V Λ = Ψl 3<rfci)> (Λ - 3«?( ϋ) Hence if f (ί) is a pure tensor of type (0,/>), it olds tat

6 ANALYTIC TENSOR AND ITS GENERALIZATION 213 In te next place, if we substitute (12) into (13), ten we get Φ L ξ υ) () * * = ^, r 3 <r f (0 > (J) - 3<ίl ( /)> ω -*- ^ "'Z-t-W 3i f(i) Hence if f ( is a pure tensor of type (#,0), ten we ave (f,. from wic, in te case wen q = 1, we find Φz I' = ~(f 9, ^/ - <PΪ dr? + ^ 3 n=-f ψl\ were denotes te operator of Lie derivative wit respect to ζ\ If ξ t J is a pure tensor of type (1, 1), ten we ave from (9) φ«*.' = Ψ: 3Γ f/ - φ r ' a, ft r + V a«9», r - f, r a, *Λ 1,2 wic is noting but ^Z ϋ m Nienuis' paper [ 3 ]. In particular, we ave Φ z δ/ = 0. ω 5. Let ξ m = ζt P '..iι J) m and η ia) = η,a) ίbq ''" bl be pure tensors of type (q,p + 1) and type (q + l,p) respectively. Ten we sall verify te following formula: (14) Φ, (ft (() ϋ) W (W ) = (Φ«U Φ ) W C4> + U ω Φ«W (δ), if /> + />' =f= 0 or ^ + ^' 4= 0. In fact, te left and side is te sum of te following six terms b Λ, Λ.r a,?>λ V r -,'

7 214 S. TACHIBANA b* = SO, <p* ~ 3r φl"*)ϊm U) %;«' " ", from wic we can easily obtain (14). If a pure tensor (or a vecter) I satisfies Φ t ξ = 0, ten we sall say tat it is a Φ-tensor (or Φ-vector). If te tensor φ t is a complex structure, ten a Φ-tensor is an analytic tensor. If φ t is a product structure i.e. an almostproduct structure suc tat its Nienuis' tensor vanises, ten a Φ-tensor is decomposable. From (14) we ave ϋ) THEOREM 1. If ξ m and η {a ) Φ) υ) are Φ-tensors 9 ten so is ξ m provided tat it is not a scalar. W (δ) 6. Let us consider two Riemannian metrics g t and φ^ wic are not necessarily positive definite. Putting <p t = φ ir g rj we sall introduce te operator Φ wic is associated to ψi. Since it olds tat g ri φ φ H = φ i5 = g r <Pi> we know tat g H is pure. Taking account of g t = φ H, we obtain Φf&i = <Pι r dr 9'» - dι9i + (d<pi r )9n + OtφΠgr = ~ 2 φ ιt [[}i1 0 - I/,} φ l were \i} g and \t\ φ are te CristoffeΓs respectively. Tus we ave symbols formed by g μ and φ H THEOREM 2. Let g Jt and φ όi be two Riemannian metrics. Ten a necessary and sufficient condition in order tat te CristoffeVs symbols coincide wit eac oter is tat Φβμ = 0, were Φ is te operator associated to φs = <p ir g r. In te rest of te present section, w τ e sall assume tat g t is a Φ-tensor, and denote by Vi Λ e operator of te Riemannian covariant derivative wit respect to g H. From Teorem 2, we know tat Φtfμ = 0 is equivalent to Vkψi = 0. Let R k3i and S ki be te Riemannian curvature tensors formed by g H and <p 3i respectively, ten we ave R ki = S ki by means of te assumption. Applying te Ricci's identity to φ. \ we get R kjr r φ? = R ki φ\ wic /ι sows tat R ki r is pure wit respect to i and. Hence R ki R ki g r is pure wit respect to k and and also pure wit respect to i and. On te oter and, S ki being te Riemannian curvature tensor formed r by φ H, if we put T ki = S ki <p r, ten we ave (15) T ki = T ik. Since it olds tat T kί = R k i r <P and T ίk = R kri φ/', te equation (15)

8 ANALYTIC TENSOR AND ITS GENERALIZATION 215 r becomes R kir φ = R kri φ[, wic sows tat R ki is pure wit respect to and. Terefore R ki is pure. Since we ave Vi ^ίλ is a ls pure. = ψΐ (V* ^HΛ + V ^r*ί Λ ) 9 /Vi ^fcfr = ~ <P (V* ^iltr + Vi ^ίkr + Vί Rkir) = 0, LEMMA 7. Let us assume tat Φ t g }i = 0. If a tensor, say T, and its covariant derivative are pure, ten we ave PROOF. Let T be a tensor of type (l, 1), for example. Ten we ave V t Φ,7V - Φ,V ( 7V = ψ[(s/ t Vr -VrVt) T t * - (v,v, On te oter and, it olds tat (VtVi - ViVt)Ti 5 = RJ Tΐ ~ Ruΐ T r = R tlr } Ψ: T t ' - R thr φ r s = R trs5 Ψι T t s s r - R tri Ψι 77 TJ From tese equations, we find tat te lemma is true. q. e. d. If we apply Lemma 7 to our R ki 9 ten we ave Φ t v* Rkt = 0, wic sows tat ViVt Rw is pure. Tus we get THEOREM 3. Let g όi and φ 5i be two Riemannian metrics and Φ be te operator associated to ψι = <p ir 9 r > If ΦiQi = 0 is valid, ten R ki and its successive covariant derivatives are pure. Let cpi 5 be an almost-product structure, ten tere exists a Riemannian metric g H suc tat g rs φ[ψi S = g im Ten we know tat te tensor φ H = φ g ri is also a Riemannian metric. Tus teorems in tis section are applicable to tis case. J 7. In tis section we sall assume tat φ φ r If we put Φ L φi J = N H, ten it olds tat = δ/. N u = φΐcϊrφ + faφπφr' + O t φ r 3 ~ dr<p L J) <pϊ

9 216 S. TACHIBANA wic is noting but te Nienuis' tensor [ 1 ], [3], [4], [10], [11], [12]. It satisfies te equations N H = - Nu 3, N : - - N r ιr φ J φ u r. J Te last equation sows tat N ιt is ybrid wit respect to i and, ence taking account of te skew-symmetricity of N ti \ it is pure wit respect to i and /. Tus we get ΛΓirVi' = JV rl VΛ N lr r = 0, N^φ? = 0, by virtue of Lemma 4 and Lemma 5. Now we introduce an aίfine connection TΊ suc tat i = 0, S i =~(ε/8)n, ι \ were v* denotes te operator of te covariant derivative wit respect to Γ, and Si 1 its torsion tensor. It is known tat tere exists suc a connection, wic will be called te canonical connection [12]. If we make use of te canonical connection, te equation (10) becomes * Z..0) Making use of te form (16), we sall obtain some formulas on te operator Φ. Te tensor ψi being pure, if we substitute it in te place of ζ or η in (14), ten we get te following formulas. We can see also tat Λ - r - + ivλ )V " *, if ςr > 1, υ) 9 / v r..ι I + N ttk rξ,,... r... 1 t (Λ, iίp^l, are valid. In te next place, we sall prove te following formula: (17) Φ, (i) <" = - W<ϊvW Λ + Σ Mr*f(*Λ- p - Λ.

10 In fact, we ave ANALYTIC TENSOR AND ITS GENERALIZATION 217 * SCO = ~ ^ Φr fw (Λ + 2 ΛΓ Ir * f «>'* ' - ' Λ q. e. d. Especially, for a pure tensor (ί) of type (0,/>), we ave (18) Φ t ί(θ=-9>i r Φrf<o, from wic it olds, taking account of (14), (19) NJ f r(<) + φΐ Φ, f r(ί) = - φϊ Φ r f*(*). From (18) we ave THEOREM 4. Let φ t satisfies φ tr φ r3 =±= δ tj. If f (/) w a Φ-tensor, ten so is ξ (ί). In tis case, we know by virtue of (19) tat te relation Nξ ίp... r... = 0 olds good. Next we sall generalize te fact tat Φiψi 3 = N ti is ybrid wit respect to / and. THEOREM 5. Let φ satisfies φ tr φ r = S Si. If ξ w is a pure tensor of type (1, p\ ten Φ z lα/ is ybrid wit respect to I and. In fact, we ave by virtue of (17) On te oter and, taking account of (14), ws find tat From tese equations we obtain te teorem. q. e. d. If we define A ιki = N ιk ani b N a b\ ten it is evidently a pure tensor of type (1,4), ence Φ t A mi is a tensor wic is ybrid wit respect to t and

11 218 S. TACHIBANA. It depends only on φ t and contains its second derivatives. From Lemma r 4, we ave Φ r A ιki = 0, wic may be a new identity on <p t. We denote by C te contraction's operator, i. e., if C means te contraction wit respect to i 1 and l9 for example, ten Cζ^ί) ϋ) = ξ ip... ( r i'" 3 - r If ξ U) m {i) is a pure tensor of type (q,p), ten te tensor Cξ U) (i) is also pure if it is not a scalar. Making use of (16), we can verify te following relation, after some calculations. (20) a) In (20), we assumed tat Cξ {i) indices of bot sides. From (20) we ave is not a scalar and C operates on te same THEOREM 6. Let φ satisfies φϊφ = δi U). If ξ {i) is a Φ-ίensor, ten υ) so is Cξ {i) provided tat it is not a scalar. be pure tensors of type (q 9 p) and type (p, q) respect- α) (i) Let (ί) and η a) ively. Ten we ave W" Φ«f (,> (Λ = vj l) ΨΪ Vr U 5) ~ Voϊ* V, L a \ because we ave from Lemma 4 and te ybridity of Nu, In te same manner, we get f (0 (Λ Φl V inw = U Λ ψΐ Vr W ~ f CO 0 ' V, L W Hence we obtain 8. Let us consider an almost-hermitian space M wose positive definite Riemannian metric is g H and te almost-complex structure is ψι. By definition tese tensors satisfy g rs φ[ψi = g^, from wic φ 5i = φf g ri is skewsymmetric. Now we assume tat Vr ψι = 0, were Vr denotes te operator of te Riemannian covariant derivative. Te following lemma is known [ 7 ], [8]. LEMMA 8. Let M be a compact almost-hermitian space satisfying Vr ΨΪ = 0. If scalar functions f and g satisfy d t f= φl d r g, ten tey are bot constant over M. From tis lemma and (21) we ave

12 ANALYTIC TENSOR AND ITS GENERALIZATION 219 THEOREM 7. Let M be a compact almost"hermitian space satisfying V*<Pi r = 0. If ξ w U) and η a) {l) are Φ-tensors of type (q,p) and of type (p, q) respectively, ten te inner product ξ^u) Vuΐ 0 is constant. COROLLARY. Let M be a compact almost-hermitian space satisfying Vrψi = 0. Ifξ V ) U) is a Φ-tensor of type (q,p) and v % {a = 1,,/>), u % {a = 1, Λ constant.,q) are Φ-vectors, ten te inner product (O α) v ip v u q u is P 1 9. Let us consider a Kalerian space M wit a positive definite metric. We sall make use of te notation in 2. An analytic tensor f(<) (Λ is by definition a pure tensor suc tat y ξ {l) a) is also pure. Now we define, for a pure tensor ζ(i) U \ ^«(0 (</) α) ( ί ) = 0 is equivalent to tat te pure tensor (o is analytic. k On taking account of tat te Riemannian curvature tensor R ki and Ricci tensor R 5i of a Kalerian space satisfy we can easily obtain v p %» ( " = vvrfo 0 ' + 2**. (.Λ" r - Λ - 2i?Λ-,...,Λ were \7 r = P rί Vί Hence if f (ί ) <Λ is analytic, ten it satisfies V r V,l.ω ω + 2iV* cov ' A - ΣΛΛ ^ = 0. Putting ( % = fvi. g g.iφ q Λi»if(o (ft) e ^ we ave, after some calculations, were (22) Tus, by Green's teorem, we ave THEOREM 8. 5) 7w a compact Kalerian space M, te integral formula 5) For a skew-symmetric contravariant pure tensor, see [6].

13 220 S TACHIBANA I r _ t U) t V 1 D Jkt JQ'.r.. _ \^ ft r t (J)\ t() M Vrb(i) -Γ ^Af 9(0 Z^^ik bip r /i JC (i; ίί valid for a pure tensor (o 0), were d<r is te volume element of M and a 2 (ξ) is given by (22). THEOREM 9. In a compact Kάlerian space, a necessary and sufficient condition for a pure tensor ζ^) U) to be analytic is tat it satisfies Σ On te oter and, in a compact orientable Riemannian space, a necessary and sufficient condition for a skew-symmetic tensor!(,-) to be armonic is tat [13] Let ζ (i) be a skew-symmetric pure tensor, ten Rc ζt p "r "S' ' :=z 0, by vritue of Lemma 5 and te ybridity Tus we ave of R k rs wit respect to r and s. COROLLARY [13]. In a compact Kάlerian space, a necessary and sufficient condition for a skew-symmetric pure tensor to be analytic is tat it is armonic. * If ξ (<) is skew-symmetric pure tensor, ten so is (0 by virtue of Lemma 2. Hence taking account of Teorem 4, in a compact Kalerian space, if a pure tensor (ί) is armonic, ten so is f (0. BIBLIOGRAPHY [ 1 ] LEGRAND, G., Sur les variete a structure de presque-produit complexe, C. R. Paris (1956), [2] LEGRAND, G., Structure presque ermitiennes au sens large, C.R.Paris (1956), [3] Nienuis, A., X 2 n-forming sets of eigenvectors, Indag. Mat. 13 (1951), [4] OBATA, M., Affine connections on manifolds wit almost complex, quaternion or Hermitian structure, Japanese Jour, of Mat. 26(1956), [ 5 ] SASAKI, S. AND YANO, K., Pseudo-analytic vectors on pseudo-kalerian manifolds, Pacific. Jour. 5(1955),

14 ANALYTIC TENSOR AND ITS GENERALIZATION 221 [6] SAWAKI, S. AND KOTO, S., On te analytic tensor in a compact Kaeler space, Jour, of te faculty of Sci. Niigata Univ. (1958), [7] TACHIBANA, S., On almost-analytic vectors in almost-kalerian manifolds, Tόoku Mat. Jour. 11(1959), [8] TACHIBANA, S., On almost-analytic vectors in certain almost-hermitian manifolds, Tooku Mat. Jour. 11 (1959), [9] WALKER, A. G., Connexions for parallel distributions in te large, Quarterly Jour. (1955), [10] WALKER, A. G., Derivation torsionnelle et seconde torsion pour une structure presquecomplexe, C. R. Paris (1957), [11] YANO, K., Te teory of Lie derivatives and its applications, Amsterdam, (1957). [12] YANG, K., On Walker differentiation in almost product or almost complex spaces, Indag. Mat. 20(1958), [13] YANO, K. AND BOCHNER, S., Curvature and Betti numbers, Annals of Mat. Studies, No. 32, (1953). OCHANOMIZU UNIVERSITY, TOKYO.