6.4. Zero powers of Riemann

Surprisingly, there are monomials of the Riemann tensor which are zero due to the cyclic symmetry only. This first happens for a single monomial of degree 6:

It does not happen for degrees 1 to 5:

In[209]:=

Cases[RInvRules[2, 1], HoldPattern[_→0]]

Out[209]=

{}

In[210]:=

Cases[RInvRules[2, 2], HoldPattern[_→0]]

Out[210]=

{}

In[211]:=

Cases[RInvRules[2, 3], HoldPattern[_→0]]

Out[211]=

{}

In[212]:=

Cases[RInvRules[2, 4], HoldPattern[_→0]]

Out[212]=

{}

In[213]:=

Cases[RInvRules[2, 5], HoldPattern[_→0]]

Out[213]=

{}

There is a single case in degree 6:

In[214]:=

Cases[RInvRules[2, 6], HoldPattern[_→0]]

Out[214]=

{I_000000,1461→0}

And 19 more cases for degree 7:

In[215]:=

Cases[RInvRules[2, 7], HoldPattern[_→0]]

Out[215]=

There is no obvious reason why this should be zero, and  for all dimensions:

In[216]:=

RInv[metric][6, 1461]//InvToRiemann//NoScalar

Out[216]=

R_ac  ^(  ef) R_    ^abcd R_b   ^( ghi) R_ (d h )^( j k) R_eij ^(   l) R_fkgl^    

but we can check it in the pure Ricci case:

In[217]:=

expr = %/.RiemannCD[a_, b_, c_, d_] →RicciCD[a, c] delta[b, d] + delta[a, c] RicciCD[b, d] - RicciCD[a, d] delta[b, c] - delta[a, d] RicciCD[b, c]

Out[217]=

In[218]:=

expr[[1]] expr[[2]]//ContractMetric//ToCanonical

Out[218]=

In[219]:=

%expr[[3]]//ContractMetric//ToCanonical

Out[219]=

In[220]:=

%expr[[4]]//ContractMetric//ToCanonical

Out[220]=

In[221]:=

%expr[[5]]//ContractMetric//ToCanonical

Out[221]=

In[222]:=

%expr[[6]]//ContractMetric//ToCanonical

Out[222]=

0

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