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 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 - 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. , } = 0. V-,:Lp In particular, if the form P(f. . ) is negative definite, then in Vn there exists no non-zero S-Killing field. 3. The basic formula Let Vn be a compact orientable Riemannian manifold of class C°° . With the help of a symmetric tensor field f. - of class C°° we form a vector field 1 ' • * p 4 iA i,,...iXo• • •Xn _ j 9= t 2 J • • • 3. _ Then the divergence of the given vector field is provided by the formula 1 2 2 J 117) j» ,j^. = f P.iJi1#i1 f\ 2*" ± p +f P.'," f i 2"**± p.'± . - 1041 - 10 M, Jaszczak Prom Eicci's identity ii2...ip ii2...ip 1 ,J,k ~ r ;k?;j " ± ai2...i P is ii2...is_1ais+1...ip = ajk f + L^ H ajk 1 s=2 we obtain ii0.»»i_ iio»««i_ f 2 P.,=f2 p,-+ 2 p + ^s „ii2,,,is-1ais+r,,ip 2^Ra3if 8=2 Prom this and from (17) we obtain l2 -1 1> + Eab/ " I'f 1 , + ao c. *# * p 3 P Dl - (p - 1) Ra;jlb f f i3...ip + lio • • • J*— «i + f 2 P f3 i2...ip;i or equivalently 1 ii ^V-- ? f3 \ _f 2-"*p fd. . + ,:L ; j i2...ipy/;i ;i;j V p + bab f 1 i ...i &abcd * 1 i,.. .i 92 n p J V Hrj***^"« -i + f 2 p. • i3- »3 ^* * * "'"p' - 1042 - On symmetric tensors 11 Next we consider a vector field f1 of the form • • • A 41 J _ f1 ' P 1f J ,1 -li 2...l1 p • Then the divergence of the vector field is given by the formula 2* * p' •f 2 p + f 2 P.fJ 1 + r r * ;i;j i2...ip ;|i i2...ip;j Prom (18) and (19) we obtain (20) fli2 ip f3 f ( "' ii i2...ip)ii " ("'"'Si V.ip);3 2 = F(f± ± J + f P. f3 + 1** p 2*** 0' Ho • • • -t - f 2 P f J ;i i2 • ••ip >j Integrating both sides of the equality (20) over the whole manifold and making use of Green's theorem we obtain ijL p r o2 « • • P »d (21) / P(f. , ) + f fi± , + n ii,,...i . 1 2 p - f ti f^ i .J dv = 0, , J. , 2' .* il p' where ii i fit2.-.ipij = f 2-" p gaj( !a - 1043 - 12 M. Jaszczak Taking into account the symmetry of the tensor field 1 * * • • rv f p with respect to all indices we obtain the identity 1 1 = _ ±1 f 1•*•' n'p J f. . . + f f^ 1 Ji~2 • • • 5 P ir..i 53 (^••.ip.j) + .£±1 f f p (ir..ip;j) From this and from (21) we finally obtain 1 i-i • • • > 0 ¡22) /[*%..! ' ' I f fi i + n f p (ir..i ¡j) 2 P -p D - f 1 dv = 0. i2...i 4. Some applications of the basic formula Let us ubserve that if a tensor field f. . is a V'^p field of S-Killing tensors, then the formula (22) takes the form 1 dv = 0. <*3) /['(flr. , ) -If' .Consequently, from the inequality F(f. ^ ) 0 it fol- lows that p = 0 - 1044 - On symmetric tensors 13 and P(f, , ) = 0. 1r,,1p Moreover, if the form ., ) is negative definite, then f. = 0. Hence in this way we have obtained ano- ther proof of Theorem 5 for the orientable manifold V . Por the S-harmonic tensor the formula (22) takes the following form r r i-i • • i F(f + f P f dv = 0 (24) / i i > i l ll'A Sr L 1" P 1"« p»«J_ Vn Prom above and from the inequality P(f. • ) > 0 we 1r,,:Lp obtain the following inequalities f. . . . = 0 • • • 1p 5«] and P(f, , ) = 0. xr'^p Moreover, when the form P(f^ 4 ) is positive definite 1" P f. = 0, we obtain in this way another proof of Theo- ir..ip rem 4 for the case of the orientable manifold Vn. 5. Necessary and suficient conditions in order that a tensor field be an S-harmonic or S-Killing tensor field For any tensor field f. . we put V p - 10 45 - 14 M.Jaszczak (25) T(f, , ) := êdk ^ i .<.}, + 1 * * * p•« > f Rai 'i i a i i + xa 1 ' ' ' s-1 ^-s+r"1? 3 = 1 - 2 y- Ra . b S , t = 1 V s < t If the tensor field f, . is an S-Killing field, ir..ip then, as is easy to see, we hav-e the identity (26) T(f, , ) = 0. ir..ip We shall show that for the compact and orientable mani- fold V the condition (26) is also sufficient in order that n the tensor field f. . be a field of S-Killing tensors. ir..ip In fact, integrating the Laplacian of the scalar function 3.4 • • • (27) / ia-V \ L 11 * * * 1n ' «5 + f P f • • . • dv = 0. 11 * * Multiplying the equality (22) with p and then adding to the result the equality (27) side by side, we obtain - 1046 - On symmetric tensors 15 (28) /,[flr"1P(± .,.k) + pP(f± ±) + J ^ 1"* * p'^' l1,,,lp (i.j... i ; j + pf 2 p • fJ. , • dv = 0. Making use of the identities (8), (9) and (25), we can trans- form the equality (28) to the following form i ,,,i (^••.ip.j) (29) c rf i p T(f ) + (p+Df( ,f - p fii2"-:iPi. f j . ... dv. 12«.«ip»3 In view of the last identity we obtain the following the- orem. Theorem 6. On the compact and orientable Riema- nnian manifold V , a symmetric tensor field is an S-Killing field if and only if we have T(f. , ) = 0, 11••*1D f1, i .4 = 0. •••iVn fl For an arbitrary tensor field f. . we put 1r ••S - 1047 - 1 6 M.Jaszczak ik (30) W(f. . ) = g f;, ± .,.k + 1'•• p»"i P - Y"* Ra f. • • + p ¿—f is iv1s-ia:Ls+r'^p S = 1 p R f P 2—, i8tB i1...ia_1aiB+1...it_1bit+r..ij' 8 , t=1 S If the tensor field f. A is an S-harmonic field, "1V1P then by definition we have w(f. * ) = o. 7,'e shall show that for the compact and orientable Riemannian manifold V the converse theorem also holds. Namely, sub- n tracting the identity (22) from the identity (27), side by side we obtain vs , - P(fi1 ,1i J + J [ ir..ip;a; * r" p 3 i-j • • • ip>Ò ) + s±l f^'-'V f< JB±1 ff. , + ir..i " p (iv..i ;j) + f 2 p _ dv = 0. From this, after some transformations, we obtain - 1048 - On symmetric tensors 17 / f3"r"iPw(f. , ) i [ 1i,,,iP n f ' P f. - f„ , f ' P + , f 2 p J + 1 .- i 1f i 1 • i dv = 0, , j. 2* * * p' " and consequently we get the identity (31) f f11"'1? W(f. , ) + i I 1Vip i,|...i ;j (i ...i f " _ f '1 t> 2 p + f ;i ig»•.i ;j dv = 0. Prom the last identity we infer that if W(f. . ) = 0, then f(. . = f± ± • and ft . m± = 0, which means that the tensor field is S-harmonic. Hence we have pro- ved the following theorem Theorem 7. On the compact and orientable Rie- mannian manifold V. a symmetric tensor field f. . n l1'"lp is an S-harmonic field if and only if W(f. , ) = 0. 1'' * p 6. Concluding remarks In [l] u.C.Chaki and B.Gupta have proved, among others, the following theorem: "A conformally symmetric Riemannian - 1049 - 18 M. Jaszczak space is a space with the constant scalar curvature if and only if the derivative of Ricci's tensor is a symmetric ten- sor". It is not difficult to show that in the case where Ricci's tensor satisfies the condition R(ij;k) = Rij;k then also the following condition holds By Definition 1 from these conditions it follows that in this case the field of Ricci's tensor is S-harmonic field. Accordingly, the theorem quoted above can be formulated equi- valent ly as follows Theorem. A conformally symmetric Riemmaniah spa- ce >3) is a space with constant scalar curvature if and only if the field of Ricci's tensor is an S-harmonic field. Prom this Theorem we immediately obtain Corollary 1. In a conformally symmetric Rie- mannian space Vn(n >3) with constant scalar curvature, the field of Ricci's tensor is an S-harmonic field. BIBLIOGRAPHY Chaki, B. Gupta: On conformally sy- metric spaces. Indian J. Math 5 (1963) 113-122. ^2 Jk. Yano, S. Bochner: Curvature and Betti numbers. Princeton - New Yersey 1953. - 105C - On symmetric tensore 19 £3 JZ. Zekanowski: On generalized symmetric cur- vature tensor fields in the Riemannian manifolds, Demon- stratio Math. 11 (1978) 157-171. INSTITUTE OP MATHEMATICS, TECHNICAL UNIVERSITY OP WARSAW Received November 22. 1977. - 1051 -