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ON HARMONIC and KILLING SYMMETRIC TENSORS in Vn DEMONSTRATIO MATHEMATICA Vol. XI No 4 197» Marian Jaszczak ON HARMONIC AND KILLING SYMMETRIC TENSORS IN Vn 1. Introduction In this paper we introduce the notion of a symmetric field of harmonic tensors and the notion of a symmetric field of Killing tensors. Similarly as in [2], we develope parallelly the theories of these both symmetric fields. It has turned out that almost all theorems known for ordinary harmonic fields can be carried over to harmonic and Killing symmetric fields. Moreover, if iri place of a Riemannian curvature ten- sor we apply a symmetric curvature tensor [3], then we also obtain a full analogy between corresponding formulas. 2. Symmetric fields of harmonic and Killing tensors in Vn Let Vn be an n-dimensional Riemannian manifold of class C°°, with a metric tensor g and a Levi-Civita connection V . By IV and BV we denote, respectively,the ring of smooth functions on V„ and the module of smooth vector n fields tangent to Vn. Further, let R denote the Riemannian curvature tensor of the manifold Vn, and S the symmetric curvature tensor of this manifold [3]. We obtain the follow- ing indentity: S(x,y)z := R(x,z)y + R(y,z)x for any x,y,z € BV. Let f:BVr be any smooth tensor field of the type (0,r) on V , and let x be any chart from the atlas of - 1033 - 2 M. Jaszczak the compact manifold V . Then it is known that the laplacian 2 of the scalar function <f> (p) = f(p), p e Vn, where ||f(p)|| denotes the square of the norm of the vector field f is gi- ven in the domain of the chart x by the following formula; / 1-.,... 5 (1 ) &<P = 2 f ' f-s 1 -i + i1,...,i jk + f fi i -i-k^ ' where f. denotes the coordinates of the field f »• • • > j-p in the chart x, and the semicolon denotes covariant diffe- rentiation. Since on a Hiemannian manifold the metric tensor g is by definition positively defined, the form (2) w(p) = f 1 r (p)f. .(p) •"•I »• • •» p>« is, at every point p e VQ, a positively defined form of the coordinates f, ^ (p) for p e V„. Hence if a ten- 11»• • • »J-p 11 sor field f satisfies, within the domain of the' chart x, a system of differential equations of the form 3k T (3) g f± i -i-k = i i i i 1• p'"' -L-]»• • •»xpJ-)»•••» Jp where T. denotes the coordinates of some ,••«,lpji»*»*>jp tensor T of type (0,2r) on Vn, and if the following inequality holds (4) T = T. f ' r f ' r >0 -*••]»•••» -Lp J i »• • •»Jp then from (1) it follows that A <j> > 0 - 1034 - On symmetric tensors 3 and consequently (by Bochner's lemma [2]) we obtain = 0 which implies f. i i = 0 and T = Tj .. a f f = 0 If the form T is positive definite, then the equality T = 0 implies f = 0. In this way we have obtained the following theorem: Theorem 1. [2]. If on the compact Riemannian r manifold VQ a tensor field f : BV - FV satisfies, within the domain of an arbitrary chart x, the conditions (3J and (4), then Vf = 0 and 1 r 1 r T = Ti-. | ,..., i, p j, 1 ,.. ., , j f f = 0. If, moreover, the form T is positive definite, then there exists no non-zero tensor field satisfying the condition of the theorem. Let f : BVr PV be an arbitrary smooth symmetric tensor field on Vn. Making use of Ricci's identity we ob- tain for f the following identities holding in the domain of an arbitrary chart x e atl Vn: jk Jk (5) -g f± ± ± .1>k + g (f± i L + fn i .± + 1 2*** D 11 2* * * * * p'D J' " 2** pD' 11 + ... + f± ^ ii2"-ip-i + - 1035 - 4 1,1. Jaszczak "" (^ 1 1 • a • 1 + f i i • a • i + ••• + 2** p 1 * 2 3. \ r—P» Q + f ;a;i = R fi #,i ai + ivVl P HS — J ^ 1 • s-1 s+1 • • ^p p - 2 y^ Ra. bf. isit ir,,is-i a is+i • •#it-i b it+r,,1p S f X — 1 3 < t or equivalent identities (6) pgdk f± i si.k - gjk(pfi i -i ~ fi i i i-i + - ... - f 1i2",ip-1 ^ip)?k + ^ "Î • B • i Î •fl • 1 + .•« + 2*'* p' ' 1 13** p' ' 2 + fa ) = N R.® f ... + i i ,,,i ja;1 i i i ai l 1 2 p-1 p *-rS —t I a V s-1 s+1"' p p " 2 R isit fiViB-1aia+1',,it-1bit+Vip" S f T — I s < t Hence we obtain the following conclusion: if a symmetric tensor field f< * satisfies the identities V'^p (7) g^ifi i -i + fn i -i + ••• + 1"* p'" " 2" pP' 1 + fiiV:Vi^V5 k ' {fV--Va;ii + + ^ a i .B.i + ••• + i a,fi."» J ? 0 ii p' J 2 ir,,ip-i ja;ip - 1036 - On symmetric tensors 5 then it satisfies the identities P Jk (3) -g f± ± J KTs^ _ _ T J"i ' ' * "r.» J » ' ' Q -i -j o-i -i 1 p s=i 8 1r ••1s-iais+r,,-p p a b - 2 \ ' R . f. I—r—' ^-sH i1,"is-1ais'-r,,lt-1blt+r ,,:Lp S j X = I set and consequently also the identities a b i i i - p (p—1 ) B . fh, , f 1 2"- P 12 p Expressing the curvature tensor R^j^l in (9) by 'the symme- tric curvature tensor we obtain ±1 812 1 + (10) -f Vf X i L .j.k-PSab'K uu "' ^! i ± 1 »• • •»- p» J » 2" * p abi cd +, p(p-U c ff 3 —s ff r abed i3...i • where Sfib = Rab [3]. Putting ai0...i b (11) F(f. ) := S , f 2 Pf , . + l1",lp aD 2* * p + H £ abiy.ip ^ + r 1 2 abed 3 p - 1037 6 M. Jaszczak we obtain from (10) and (11) the following identity: /-r-'Sgjkf = _pP(f 1*** p'"' 1*** p Theorem 2. If on the compact Riemannian mani- fold V a symmetric tensor field fj j satisfies the n 1r"1n identity (7) and the inequality P(f. ] ^ 0, then the 1 • • •P n tensor field f is covariantly constant fi i -1 = 0 and consequently F(f, , ) = 0. ^•r ,,:lP If the form F(f. ) is negative definite, then there ir..ip exists no non-zero symmetric tensor field satisfying the identity (7). Analogously from (6) we infer that: if a symmetric tensor field f. satisfies the identity (12) g^ipfi i -i " i i i-i -fi i in - - 1** p'" 2 3** * p ' 1 1 3** * p"' 2 f + fS + " i1i2...i ( i2...i sas^ + f& + + ] 0 ±l il ,,,li «a-,a il ••• i i -a-i = 1 3 p ' 2 12** p-1' ' p then it also satisfies the identity P jk R& f (13) pg f± i -i-k = F i " i ai i l1***lp'«J»K !—* S 1 * * S-1 S+1 * " p S=J - 10.38 - On symmetric tensors 7 P - 2 y^ Ra. b f. isit ivis-iais+r ••it-ibit+r,,iP s < 1 and consequently also the identities /v-ip^k i.i.k = P(fi i ), 1••• p»«• 1** p where P(f. ) is defined by (11). 1''' p Now from Theorem 1 we obtain following theorem. Theorem 3, If on the compact Riemannian mani- fold V a symmetric tensor field f. satisfies the n ,...,ip identity (12) and the inequality P(fi i i ) > 0 12* * p then it is a covariantly constant field, i.e. fi i -i = 0 and consequently the following equality holds F(f, , ) = 0. H* *#1p If the form P(f. ^ ) is positive definite, then there 1V1p exists no non-zero symmetric tensor field satisfying the identity (12). Now we introduce the following definition. Definition 1. A tensor field f will be called a symmetric field of harmonic tensors, shortly: S-har- monic field, if within the domain of an arbitrary chart x e atl Vn„ it satisfies the identities - 1039 - 8 M. Jaszczak (14) j V = flr-^ -p5 [15) •p - o 1 i -a - * 2* * * p' Definition 2. A tensor field f. will be called a symmetric field of Killing tensors, .shortly S-Killing field, whenever within the domain of an arbitrary chart x g atl. it satisfies the following identity I(i1...ip) ^...ip (16) f(ir..ip;d) = 0 fa. • • •. #) Q-= 0. Prom the definitions above it follows that an S-harmonic field satisfies the identity (12). Wow from Theorem 3 we obtain the following theorem. T h e cf r e m 4. If on the compact Hiemannian mani- fold V an ¿.-harmonic field f. satisfies the ine- H l«t • • • i quality 1 p F (f • . ) > 0 ir..ip then this field is covariantly constant, i.e. f. = 0 11* * *1p'^ and consequently P(f, , ) = 0. i1... - 1040 - On symmetric tensors 9 In particular, if the form P(f. ) is positive definite, 1' *' p then Vn does not admit an S-harmonic field different from 0. Analagously, an S-Killing tensor field satisfies the iden- tity (7), hence by Theorem 2 we obtain the following theorem Theorem 5. If on the compact Riemannian mani- fold V an S-Killing tensor field satisfies the inequality F(f, , ) « 0 then this field is covariantly constant, that is, f i 1 * * i* -p'*i '= °> and consequently F(f.
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