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/* 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)))$
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