DEMONSTRATE MATHEMATICA

Vol. XI No 1 1»7I

Dedicated to Profeiaor

Stefan Strasxewicz

Zbigniew ¿ekanowski

ON GENERALIZED SYMMETRIC CURVATURE FIELDS IN THE RIEMANIAN

1. Some properties of the 3 and 4-linear maps between the mod ale a Let P be a linear ring, V and W the P-modules. It is not difficult to prove the following lemma [2J. Lemma 1. If L:V^ —W is a P-4- of the P-module V into the P-module W satisfying the iden- tities

L(x,y,z,u) = L(x,z,y,u) = -L(x,y,u,z) for all x,y,z,u e V, then L = 0. We shall prove the following lemma. Lemma 2. If S : V^ —- W is a P-4-linear map of the P-module V into the P-module W satisfying the iden- tities

(1) S(x,y,z,u) = S(y,x,z,u) = S(x,y,ufz) and

(2) S(x,y,z,u) + S(y,zfx,u) + S(z,x,y,u) = 0 for all x,y,z,u e V then the map S satisfies the identity

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S(x,y,z,u) = S(z,u,x,y) for all x,y,z,u e V. Proof. Let the map S satisfy the assumptions of Lemma 2. Then S satisfies the identities

(3) S(u,y,z,x) + S(y,ztu,x) + S(z,u,y,x) = 0 for any x,y,z,u e V. Prom (2) and (3) it follows that

(4) S(x,y,z,u) - S(ztu,x,y) = S(y,u,ztx) - S(z,x,y,u) for any x,y,z,u e V. Now, let us put

(5) D(x,y,z,u) = S(x,y,z,u) - S(z,u,x,y) for all x,y,z,u e V. The map D t V* W defined by the formula (5) satisfies the identities

D(x,y,z,u) • D(y,x,z,u) = - D(x,z,y,u) for all x,y,z,u 6 V. Hence by virtue of the lemma 1 D «= 0. Consequently we obtain

S(x,y,z,u) = S(z,u,x,y) for any x,y,z,u c V. o Let cj i V—» P be an antisymmetric linear map of the P-module V into the ring P. It is not diffioult to see that the map S i V^—P defined by the formula

S(x,y,s,u) - oj(x,u).w(y,z) + cj(x,a )*w(y,u) for all x,y,z,u e V satisfies the assumption of the lemma 2.

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How let us consider a P-4-linear map A : V4 W of the P-module V into the P-module W satisfying the iden- tities

A(x,y,z,u) = - A(y,x,z,u) = - A(x,y,u,z) = A(z,u,ac,y) for any x,y,z,u e V. It is easy to verify tjhat the map S j V4—-W defined by the formula

S(x,y,z,u) = A(x,z,y,u) + A(x,u,y,z) for any x,y,z,u e V satisfies the assumptions of the lemma 2 also* Now, if S : V4—-W is a P-4-linear map of the P-mo- dule V into the P-module W satisfying the identities

S(x,y,z,u) = S(y,x,z,u) = S(x,y,u,z) = S(z,u,x,y) for all x,y,z,u e V then, as ia easy to show, the map A : V4 —W defined by the formula

A(x,y,z,u) = S(x,z,y,u) - S(x,u,y,z) for any x,y,z,u e V, satisfies the identities

(6) A(x,y,z,u) = -A(y,x,z,u) = -A(x,y,u,z) and

(7) A(x,y,z,u) + A(y,z,x,u) + A(z,x,y,u) = 0 for all x,y,z,u £ V.

Let us denote by LgB(V,W) the set of all P-4-linear maps of the P-module V into the P-module W satisfying 4 the identities of the form (1) and (2), and by L B(V,W)

- 159 - 4 Z.iekanowski the set of all P-4-linear maps of the P-module V into the P-module W satisfying the identities of the form (6) and (7).

Evidently, each of the set L^fv.W) and LgB(V,W) can be equiped with the structure of the P-module, We shall prove the following theorem.

Theorem 1. The map CP : L£B(V,W)—• L£B(V,W) defined by the formula

(8) (cp(A)) (x,y ,z,u) = S(x,y,z,u) = A(x,z,y,u) + A(x,u,y,z)

for all A e IigB(V,W) and x,y,z,u e V is an isomorphism of the P-modula Lj^V.W) onto the P-module L^iV.W). Proof. The fact that the map cp is a homomorphism is evident. Now, we shall prove that the map is a monomorphism.

For that purpose assume that there exist A^,A2 e L^B(V,W) such that cp{A^ J = cp(A2), or equivalently that

(9) A1 (x,z,y,u) + A.j (x,u,y,z ) = A2(x,z,y,u) + A2(x,u,y,z) for all x,y,u,u £ V. This implies the identities

(10) A^(x,z,y,u) - A2(x,zty,uJ = A2Ix,u,y,z) -Ag(x,u,y,z).

Putting

(11) D(x,z,y,u) =A1(x,z,y,u) - A2(x,z,y,u) for all x,y,z,u e V, we get the map D : V^—— W which satisfies the identities by virtue of (10)

D(x,z,y,u) = -D(z,x,y,u) = -D(x,z,u,y) = -D(y,z,x,u) for all x,y,z,u e V. This shows that the map D is anti- symmetric with respect to all variables. From here and from the assumption that the maps satisfy the identities of the

- 160 - Symmetric curvature tensor fields 5 form (7) it follows that D = 0 and consequently A^ = — A P • Finally, it remains to show that the map cp is an epimor- phism. Let S be an arbitrary element of LgB(V,W). Of course, the map S satisfies the identities

(12) 3S(x,y,z,u) = S(x,y,z,u) + S(x,y,z,u) + S(x,y,z,u).

By assumption the map S satisfies the identities

(13) S(x,y,z,u) + S(y,z,x,u) + S(z,x,y,u) = 0.

Prom (12) and (13) we obtain

3S(x,y,z,u) = S(x,y,z,u) - S(z,y,x,u) + S(y,x,z,u) - S(z,x,y,u).

Now, putting

(15) 3A(x,y,z,u) = S(x,y,z,u) - S(x,u,z,y) for all x,y,z,u e V we get the map A : V^—»- W, which belongs to the P-module L^B(V,W). Prom (14) and (15) we obtain finally

S(x,y,z,u) = A(x,z,y,u) + A(x,u,y,z) for any x,y,z,u e V. Analogically we prove

Theorem 2. The map y : L£b(V,W) —- L^g(V,W) defined by the formula

(if(S)) (x,y ,z,u) = A(x,y,z,u) = S(x,z,y,u) - S(x,u,y,z)

for all S e LgB(V,W) and x,y,z,u £ V is an isomorphism of the P-module L£B(V,W) onto the P-module L£B(V,W).

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Prom the proof of the theorem 1 it follows that if (cj>(A)) (x,y,z,u) = S(x,y,z,u) = A(x,z,y,u) + A(x,u,y,z) then

••1 1 (cp (S) )(x,y,z,u) = A(x,y,z,u) = -^(Six.z.y ,u) - S(x,u,y,z))

for any x,y,z,u e V.

Now, let ub denote lay lJB(V,W) the set of all P-3-li- near maps of the P-module V into the P-module W satisfying the identities

^(x,y,z) = ^(y,x,z) and

^(x,y,z) + ^(z,x,y) + f{ y,z,x) = 0 for pe L^b(V,W) and all x,y,z e V.

Let us denote also by L^B(V,W) the Bet of all P-3-li- near maps of the P-module V into the P-module W satisfy- ings the identities

K(x,y,z) = -K(y,x,z)

and

K(x,y,z) + K(z,x,y} + K(y,z,x) = 0 for K e ^(V.W) and all x,y,z e V.

Evidently, each of the sets I,^B(V,W) and L^B(V,W) can be equiped with the structure of the P-module. Analogically we prove too

Theorem 3. The map cp : I

((x,y,z) = K(x,z,y) + K(y,z,x)

for any K € L^B(V,W) and x,y,z e V is an isomorphism of

the P-module I

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7

Theorem 4. The map y : L^g(V,W) —» L^B(V,W) defined by the formula

(iplp) )lx,y,z) = K(x,y,z) =f(x, z,y) z,x) for any ff> t LggiVjW) and x,y,z e V is an isomorphism of the P-module I^B(V,W) onto the P-module L^B(V,W). We can show too that if

(

(cp~1 {f))(x,y,z) = K(x,y,z) = ^ (#>(x,z,y) -^(y.z.x))

for all x,yfz £ V. Now we prove Theorem 5. If S=V2 —- R, a = 1,2, are a. symmetric 2-lineal maps of the real V into R satisfying the identities

(16) S(x,y)«S(z,u) + S(y,z).S(x,u) + S(z,x)»S(y,u) = 0 12 12 12 for all x,y,z,u e V then S = 0 or S = 0. 12' Proof. Assume that S = V2—^ R, OL = 1,2 satisfy the assumption of the theorem. Putting x=y=z in(l6) we obtain

S(x,x) • S(x,u) = 0 1 2 for all x,u € V. This implies S * 0 or S = 0. 1 2 It is not difficult to prove Lemma 3. If S : V* —»- W is a P-4-lineal map of the P-module V into the P-module W satisfying the identi- ties of the form (1) and (2j and the identities - 163 - 8 Z.Zekanowski

S(x,y,y,x) = 0 hold for all x,y € V, then S = 0. Lemma 3 implies the following corollary, corollary 1. If S : V4 —- W, a = 1,2, are P-4-linear maps of the P-module V into the P-module W satisfying the identities

(17) S(x,y,z,u) = S(y,x,z,u) = S(x,y,u,z), a= 1,2 and

(18) S(x,y,z,u) + S(z,x,y,u) + S(y,z,x,u) = 1 1 1 = S(x,y,z,u) + S

S(x,y,x,y) = S(x,y,x,y) 1 2 for all x,y,z,u e V, then S = S. 1 2 How, let S : V4—- W be P-4-linear map of the P-module V into P-module W satisfying the identities of the form (1) and (2). Then from (2) it follows that for any x,y e V we have the identities

S(x,x,x,y) = 0 and S(x,x,y,y) » 2S(x,y,x,y).

Hence we p>et

Corollary 2. If the vectors e^ and e2 are a of the P-module V and g : V4—- W, a» 1,2 are P-4-linear maps of the P-module V into the P-module W sa- tisfying the identities (17) and (18) as well as the iden- tities

S(e^,62fe^p&2) * ®f®2*®2^ then S * S. 1 2 - 164 - Symmetric curvature tensor fields 9

In particular we have Corollary 3. IfVis 2-dimensional real vector spaoe, w skew-symmetric bilinear form on V and S : V^ B is a 4-linear map of the vector spaoe V into R satisfying the identities of the form (1) and (2) than there exists only one number x e R suoh that

S(x,y,z,u) = x[w(x,z)*

S(x,y,z,u) = S(y,x,z,u) = S(x,y,u,z) and

S(x,y,z,u) + S(z,x,y,u) + S(y,z,x,u) = 0 for all x,y,z,u t V there is a function

correotly defined by the formula

(19) ka(er) s(x,ytxty) g(x,x)g(y,y)-(g(x,y)) for any ff e G^, where x,y are arbitrary linear independent vectors of the plane (> . It is easy to show, by a simple calculation, that the function k0 indeed does not depend upon the vectors x and y determining the plane ff t Gg. Prom the definition of the function and Corollary 1 we obtain the following corollary.

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4 Corollary 4. If (St t V R, = 1,2, are 4-linear naps of the real vector space V into R sa- tisfying the identities of the form (18) and (17) ami if k " k , then S = S. 8 1 ? 12 1 ^

2. Symmetric curvature in Riemannian manifolds

Let Mn be an n-dimensional Riemannian of class C~ with a g and the Levi-Civita connection V . FM and BM denote, respectively, the ring of smooth functions on M, and the Fin-module of smooth vector fields on H.

It is known [1] that any map K » BM * BM —•BndpM(Blll|) satisfying conditions

K(x,y) = -K(y,x)

K(x,y)z + K(z,x)iy + K(y,z)x =» 0 and g(K(x,y)z,u) = -g(K(x,y)u,z) for all x,y,z,u e BM is called a generalized ourvature ten- sor in the Riemannian manifold Mn> Now, we introduce the following definition»

Definition 1. Any map S : BM * BM—Endpiff(BM) satisfying the conditions

S(x,y) « S(y,s)

S(x,y)z + S(z,x)y + S(y,z)x = 0 and

g(S(x,y)z,u) » g(S(x,y)u,z) for all x,y,z,u e BM is called a generalized symmetric cur- vature tensor in the Riemannian manifold Mn.

- 166 - Symmetric curvature tensor fields 11

Let us denote by L(K,M) the FM-module of all generali- zed curvature tensors in the Riemannian manifold Mn and by L(S,M) - the FM-module of all generalized symmetrio ourvature tensors in a Mn. T-heorem 3 implies the following corollary. Corollary 5. The FM-modules L(K,M) and L(S,M) are isomorphic. Moreover from the theorem 3 it follows that the map

Cp s L(K,M) — L(S,M) defined by the formula

(*) (cp(K))(x,y,z) = S(x,y )z = K(x,z)y + K{y,z)x for all x,y,z e BM and K e L(K,M) is an isomorphism of the FM-modale L(K,M) onto the FM-module L(S,M). The inverse of the isomorphism cp is defined then by the formula

(cp-1 (S)}(x,y,z) = K(x,y)z = -j(S(x,z)y - S(y,z)x).

Analogically as for the generalized curvature tensor we define the Ricci-tensor and a scalar curvature of the symme- tric generalized curvature tensors. Definition 2. For an arbitrary S £ L(S,M) the tensor denoted by Ric S and defined by the formula

Ric S(x,y) = trz S(z,x)y for all x,y,z e BM is called Ricci's tensor of the ten- sor S. How let us assume that n » 2 and denote by Gp, p e MQ the Grassmannian manifold of all 2-dimensional subspace of the tangent space M . Then the real number

- 167 - 12 Z.fcekanowski

k (O m g(S(vtu)utv) 8 g(u,u) g(v,vMg(u,v)r where Cf e Gp and v,u are a linear independent veotors of f , is called the symmetric sectional curvature for 5,

let e^,...ten be the orthonormal basis of the tangent spaoe Up, p e MQ« The real number

n 8a(p) » ^ ' Rio S^ e^)

is called the symmetric scalar curvature of the space MQ in the point p e Mn. Now, let K e L(K,M) be an arbitrary generalized curva- ture tensor of the Mfl and S e L(S,M) the image of K under the isomorphism (p defined by the formula (*). From the definitions of Rlcci's tensors for K and S and {*) we obtain the following corollary. Corollary 6. Por any K 6 L(K,M) and S - = (j)(X) € L(S,M), where (p is an isomorphism between FM-modu- les L(K,M) and L(S,M) defined by the formula (*) we have

Rie S = Rie K.

From the corollary 6 and the definitions of the soalar curvature for K and S we get Corollary 7. For any K e L

S are the same, i.e. sg(p) = Sg(p); p e Next, the definitions of the sectional curvature imply Corollary 8. For any K e L(K,M) and S = = q>(K) 6 L(S,M), where

166 - Symmetric curvature tensor fields 13

k i6) g(S(v,w)w,v) &(K(v,w)w,v) s ^ g(v,v)g(w,w)-(g(v,w)r g(v,v)g(w,w)-(g(v,w)r

Evidently, if K e L(K,M) and K = 0 then cp(K) = 0 and conversely if S e L(S,M) and S = 0 then cp~1(S) = 0. It is easy to prove the following theorem. Theorem 6, If a tensor K € L(K,M) is a cova- riant constant tensor then the tensor cp (K) e L(S,M), where cp is an isomorphism between L(K,M) and L(S,M) is a co- variant constant tensor too. Let uj : W*BM—- FM be an arbitrary skew-symmetric tensor on the Riemannian manifold MQ. How, having the we are able to define by the formula

(20) S(x,y,z,u) = oo(x,z) (y,u) + id(x,u)w(y,z) for all x,y,z,u e BM, the generalised symmetric curvature tensor field on M^. Consequently, further we can define the generalized curvature tensor field K by the formula

(21) K(x,y,z,u) = S(x,z,y,u) - S(x,u,y,z) for any x,y,z,u e BM. Similarly if S : BM * BM —- PM is em arbitrary symme- tric tensor field on Riemannian manifold M„n then we can define on Mn the generalized curvature tensoz field K by the formula

K(x,y,z,u) = S(x,z)S(y,u) - S(x,u)S(y,z) for all x,y,z,u € BM and consequently further we can define also the generalised symmetric tensor field S by the formu- la

S(x,y,z,u) • K(x,z,y,u) + K(x,u,;,z) for any x,y,z,u e BM. - 169 - 14 Z.2ekanowski

Henoe ve have Corollary 9. Any symmetrio and skew symmetric tensor field of type (0,2) on the Riemannian space Mfl de- termines in a natural way some generalized curvature tensor field and some generalized symmetric curvature tensor field by the formula.8 (20) and (21) respectively. Intuitionally it is evident that any field in the phy- sical space foroes some "ourvature" of this space. Consequent- ly, any suoh tensor field ought to determine in one way some curvature tensor field of this space allowing to measure "cur- vatures" of the considered spaoe in which the field exists. Moreover, if in this space we introduce a metric tensor in a reasonable way then in this space we will be able to con- sider the soalar curvature, the seotional curvature and the Ricci's curvature of the space. Example . Let M be an elastic isotropic medium and g a metric tensor of the space - medium M. It is well known f£om the theory of elasticity that the physical proper- ties of M are desoribed by the tensor £ defined by the formula E(x,y,z,u) • Aig(x,y)g(z,u) + /i(g(x,u)g(y,z) + + g(x,z)g(y,u)) where \ , ¡x are Lame's constants and x,y,z,u are veotors. The tensor E satisfies the identities

E(x,y,z,u) = E(y,x,z,u) = E(x,y,u,z) = E(z,u,x,y) for all x,y,z,u. This tensor allows us to define some generalized curvatu- re tensor K of the space-medium M by the formula

(22) K(x,y,z,u) = E(x,z,y,u) - E(x,u,y,z).

The tensor K defined by the formula (22) may be oalled the curvature tensor of the space-medium M. From the definitions of the tensors E and K by simple calculation we get

- 170 - Symmetric curvature tensor fields 15

K(x,y,z,u) = at(g(x,z)g(ytu) - g(x,u)g(y,z)), where x = X - fi. Henoe we obtain Corollary 10. The elastic isotropic medium U is a space of constant curvature - deformation <£ = X -p..

REFERENCES

[1] K. N o m i z u : On the decomposition of generalized curvature tensor fields, in honor of K.Yano. Kinokunigua, Tokyo 1972 ,.335-345. [2] Z. i e k a n 0 w s k i : On certain generalization of the concept of a self-dual tensor, a harmonic tensor and a Killing tensor in a Vn. Demonstratio Math. 1 (1969) 77 - 136.

INSTITUTE OP , TECHNICAL UNIVERSITY OP WARSAW Received May 30, 1977.

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