/* The second covariant derivative of the covariant Ricci Tensor */

RICCICOV(II,JJ,KK,LL):=BLOCK(MODEDECLARE([II,JJ,KK,LL,AA,BB],FIXNUM), 
IF DIAGMETRIC THEN H[II,JJ,KK,LL]:H[JJ,II,KK,LL]:
  RATSIMP(SUM(
   MCS[AA,II,JJ]*LR[JJ,JJ]*MCS[KK,LL,AA]+
    MCS[AA,JJ,II]*LR[II,II]*MCS[KK,LL,AA]+
     LR[AA,AA]*MCS[II,KK,AA]*MCS[JJ,LL,AA]+
      MCS[AA,KK,II]*LR[II,II]*MCS[JJ,LL,AA]+
       LR[AA,AA]*MCS[II,LL,AA]*MCS[JJ,KK,AA]+
        MCS[AA,KK,JJ]*LR[JJ,JJ]*MCS[II,LL,AA]-
         DIFF(LR[II,JJ],OMEGA[AA],1)*MCS[KK,LL,AA],AA,1,DIM)

-(DIFF(LR[II,II],OMEGA[KK],1)*MCS[JJ,LL,II]+
   LR[II,II]*DIFF(MCS[JJ,KK,II],OMEGA[LL],1)+
    DIFF(LR[II,II],OMEGA[LL],1)*MCS[JJ,KK,II]+
     DIFF(LR[JJ,JJ],OMEGA[KK],1)*MCS[II,LL,JJ]+
      LR[JJ,JJ]*DIFF(MCS[II,KK,JJ],OMEGA[LL],1)+
      DIFF(LR[JJ,JJ],OMEGA[LL],1)*MCS[II,KK,JJ])

+DIFF(DIFF(LR[II,JJ],OMEGA[KK],1),OMEGA[LL],1)) 

ELSE  H[II,JJ,KK,LL]:H[JJ,II,KK,LL]:
RATSIMP(SUM(SUM(
  MCS[AA,II,BB]*LR[BB,JJ]*MCS[KK,LL,AA]+
   MCS[AA,JJ,BB]*LR[BB,II]*MCS[KK,LL,AA]+
    LR[AA,BB]*MCS[II,KK,AA]*MCS[JJ,LL,BB]+
     MCS[AA,KK,BB]*LR[BB,II]*MCS[JJ,LL,AA]+
      LR[AA,BB]*MCS[II,LL,AA]*MCS[JJ,KK,BB]+
        MCS[AA,KK,BB]*LR[BB,JJ]*MCS[II,LL,AA],AA,1,DIM),BB,1,DIM)

- SUM(
    DIFF(LR[II,JJ],OMEGA[AA],1)*MCS[KK,LL,AA]+
     DIFF(LR[AA,II],OMEGA[KK],1)*MCS[JJ,LL,AA]+
      LR[AA,II]*DIFF(MCS[JJ,KK,AA],OMEGA[LL],1)+
       DIFF(LR[AA,II],OMEGA[LL],1)*MCS[JJ,KK,AA]+
        DIFF(LR[AA,JJ],OMEGA[KK],1)*MCS[II,LL,AA]+
         LR[AA,JJ]*DIFF(MCS[II,KK,AA],OMEGA[LL],1)+
     	  DIFF(LR[AA,JJ],OMEGA[LL],1)*MCS[II,KK,AA],AA,1,DIM)

+DIFF(DIFF(LR[II,JJ],OMEGA[KK],1),OMEGA[LL],1)))$
