DEMONSTRATE MATHEMATICA Vol. XI No 1 1»7I Dedicated to Profeiaor Stefan Strasxewicz Zbigniew ¿ekanowski ON GENERALIZED SYMMETRIC CURVATURE TENSOR FIELDS IN THE RIEMANIAN MANIFOLDS 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-linear map 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 - 157 - 2 Z.fcekanowski 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. - 158 - Symmetric curvature tensor fields 3 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). - 161 - 6 Z.iekanowski 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<aB(V,W) —- L^g(V,W) defined by the formula (<f( K))(x,y,z) = #>(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<aB(V,W) onto the P-module I^B(V,W).