Appendix
1.1 Proof of the Equalities \(\mathcal {A}_k=\mathcal {B}_k\)
In the following computation, the components of the Ricci tensor \({\mathrm {Ric}}\) are denoted by \(R_{ij}\). Here we give some useful formulas for the computation of the \(\mathcal {A}_k\)’s:
$$\begin{aligned} \nabla _i R_{ij}&= \frac{1}{2}\nabla _j \mathrm {scal\,}\end{aligned}$$
(28)
$$\begin{aligned} \nabla _a R_{iabj}&= -\nabla _b R_{ij}+\nabla _j R_{bi} (\text {by}\, 2^{\mathrm{nd}}\, \text {Bianchi})\end{aligned}$$
(29)
$$\begin{aligned} \nabla _b R_{iabj}&= -\nabla _a R_{ij}+\nabla _i R_{aj}\end{aligned}$$
(30)
$$\begin{aligned} \Delta R_{iabj}&= \nabla _{bi} R_{aj}+\nabla _{aj} R_{bi}-\nabla _{ij} R_{ab}-\nabla _{ab} R_{ij}. (\text {by}\, 2^{\mathrm{nd}}\, \text {Bianchi})\end{aligned}$$
(31)
$$\begin{aligned} \mathcal {A}&:= \nabla _c R_{iabj}\cdot \nabla _c R_{ibaj}=-\nabla _c R_{iabj}\cdot (\nabla _c R_{baij}+\nabla _c R_{aibj})\nonumber \\&=|\nabla {\mathrm {Riem}}|^2-\mathcal {A},\quad \text {yielding } \mathcal {A}=\frac{1}{2}|\nabla {\mathrm {Riem}}|^2.\end{aligned}$$
(32)
$$\begin{aligned} \mathcal {B}&:= \nabla _c R_{iabj}\cdot \nabla _b R_{iacj}=-\nabla _c R_{iabj}\cdot (\nabla _j R_{iabc}+\nabla _c R_{iajb})\nonumber \\&= |\nabla {\mathrm {Riem}}|^2-\mathcal {B},\quad \text {yielding } \mathcal {B}=\frac{1}{2}|\nabla {\mathrm {Riem}}|^2. \end{aligned}$$
(33)
$$\begin{aligned} \mathcal {A}_1&:= \mathcal {T}^\omega .\\ \mathcal {A}_{2}&:=-{\mathrm {Tr\,}}\mathrm {Sym}\nabla _{I'c} R_{ij}\nabla _{J'b} R_{ibcj}\overset{(29)}{=}{\mathrm {Tr\,}}\mathrm {Sym}\nabla _{I'c}R_{ij}\nabla _{J'}(\nabla _c R_{ij}-\nabla _j R_{ci})\\&=\mathcal {T}^{\omega -1}-~\mathcal {M}^{\omega -1}.\\ \mathcal {A}_{3}&:= -{\mathrm {Tr\,}}\mathrm {Sym}\nabla _I R_{ij}\nabla _{J''bc} R_{ibcj}\overset{(30)}{=} -{\mathrm {Tr\,}}\mathrm {Sym}\nabla _I R_{ij}\nabla _{J''b}(-\nabla _b R_{ij}+\nabla _i R_{bj})\\&=~\mathcal {T}^{\omega -2}_{1}.\\ \mathcal {A}_{4}&=-{\mathrm {Tr\,}}\mathrm {Sym}\nabla _{I'b}R_{ij}\nabla _{J'c} R_{ibcj} \overset{(30)}{=}{\mathrm {Tr\,}}\mathrm {Sym}\nabla _{I'b} R_{ij}\nabla _{J'} (\nabla _b R_{ij}-\nabla _i R_{bj})=\mathcal {A}_{2}.\\ \mathcal {A}_{6}&:= \mathcal {R}^\omega .\\ \mathcal {A}_{7}&:= {\mathrm {Tr\,}}{\mathrm {Sym}}\nabla _{I'b}R_{iabj}\nabla _{J'c}R_{iacj} \overset{(30)}{=}2(\mathcal {T}^{\omega -1}-\mathcal {M}^{\omega -1}).\\ \mathcal {A}_{8}&:= {\mathrm {Tr\,}}{\mathrm {Sym}}\nabla _{I''cb}R_{iabj}\nabla _{J}R_{iacj} \overset{(30)}{=}2\mathcal {B}_5.\\ \mathcal {A}_{9}&:= {\mathrm {Tr\,}}{\mathrm {Sym}}\nabla _{I'c}R_{iabj}\nabla _{J'b}R_{iacj} \overset{(33)}{=}\frac{1}{2}\mathcal {R}^{\omega -1}.\\ \mathcal {A}_{10}&:= {\mathrm {Tr\,}}{\mathrm {Sym}}\nabla _{I}R_{iabj}\nabla _{J}R_{ibaj} \overset{(32)}{=}\frac{1}{2}\mathcal {R}^\omega .\\ \mathcal {A}_{11}&:= {\mathrm {Tr\,}}{\mathrm {Sym}}\nabla _{I'b}R_{iabj}\nabla _{J'c}R_{icaj} \overset{(30)}{=} \mathcal {B}_2.\\ \mathcal {A}_{12}&:= {\mathrm {Tr\,}}{\mathrm {Sym}}\nabla _{I''bc}R_{iabj}\nabla _{J}R_{icaj} \overset{(30)}{=} \mathcal {B}_5.\\ \mathcal {A}_{13}&:= {\mathrm {Tr\,}}{\mathrm {Sym}}\nabla _{I'c}R_{iabj}\nabla _{J'b}R_{icaj} ={\mathrm {Tr\,}}{\mathrm {Sym}}\nabla _{I'}(\nabla _aR_{icbj}-\nabla _iR_{acbj}) \nabla _{J'b}R_{icaj}\\&=\mathcal {A}_9-\mathcal {A}_{13}. \hbox { Hence } \mathcal {A}_{13}=\frac{1}{4}\mathcal {R}^{\omega -1}.\\ \mathcal {A}_{14}&{:=}{\mathrm {Tr\,}}{\mathrm {Sym}}\nabla _{I}R_{iabj}\nabla _{J''bc}R_{icaj}\overset{(29)}{=}\mathcal {B}_5.\\ \mathcal {A}_{15}&{:=}{\mathrm {Tr\,}}{\mathrm {Sym}}\nabla _{I}R_{iabj}\nabla _{J''ac}R_{icbj} \overset{(29)}{=}2\mathcal {B}_5.\\ \mathcal {A}_{16}&{:=}{\mathrm {Tr\,}}{\mathrm {Sym}}\nabla _{I'c}R_{iabj}\nabla _{J'a}R_{icbj} \overset{(33)}{=}\frac{1}{2}\mathcal {R}^{\omega -1}.\\ \mathcal {A}_{17}&{:=}{\mathrm {Tr\,}}{\mathrm {Sym}}\nabla _{I'a}R_{iabj}\nabla _{J'c}R_{icbj} \overset{(29)}{=}2(\mathcal {T}^{\omega -1}-\mathcal {M}^{\omega -1}).\\ \mathcal {A}_{18}&{:=}{\mathrm {Tr\,}}\mathrm {Sym}\nabla _{I'''ab} R_{iabj} \nabla _{J''cd} R_{icdj} \overset{(29)}{=}{\mathrm {Tr\,}}\mathrm {Sym}\nabla _{I''}\Delta R_{ij} \nabla _{J''}\Delta R_{ij}=\mathcal {T}^{\omega -2}_{1,1}.\\ \mathcal {A}_{20}&{:=}{\mathrm {Tr\,}}\mathrm {Sym}\nabla _{I'''abc}R_{iabj}\nabla _{J'd} R_{icdj} \overset{(30)}{=}{\mathrm {Tr\,}}\mathrm {Sym}\nabla _{I'''c}\Delta R_{ij} \nabla _{J'} (\nabla _c R_{ij}-\nabla _i R_{cj})\\&= \mathcal {T}^{\omega -3}_{1}-\mathcal {M}^{\omega -3}_{1}.\\ \mathcal {A}_{21}&{:=}{\mathrm {Tr\,}}\mathrm {Sym}\nabla _{I'''dab} R_{iabj}\nabla _{J'c} R_{icdj} \overset{(29)}{=}{\mathrm {Tr\,}}\mathrm {Sym}\nabla _{I'''d}\Delta R_{ij} \nabla _{J'} (\nabla _d R_{ij}-\nabla _j R_{di})\\&=~\mathcal {T}^{\omega -3}_{1}-\mathcal {M}^{\omega -3}_{1}.\\ \mathcal {A}_{23}&{:=}\nabla _{I'd} \Delta R_{iabj} \nabla _{J'''abc}R_{icdj}\overset{(29)}{=} -\nabla _{I'd} R_{iabj} \nabla _{J'''dab} R_{ij}\overset{(30)}{=}\mathcal {A}_{22}.\\ \mathcal {A}_{24}&{:=}{\mathrm {Tr\,}}\mathrm {Sym}\nabla _{I''ab} R_{icdj}\nabla _{J''cd}R_{iabj}\!=\!{\mathrm {Tr\,}}\mathrm {Sym}\nabla _{I''ac}R_{ibdj} \nabla _{J''cd}R_{iabj}\overset{(33)}{=} \frac{1}{2}\mathcal {R}^{\omega -2}.\\ \mathcal {A}_{25}&{:=}{\mathrm {Tr\,}}\mathrm {Sym}\nabla _{I''da} R_{iabj}\nabla _{J''bc}R_{icdj} \overset{(29)}{=}\mathcal {T}^{\omega -2}-\mathcal {M}^{\omega -2}.\\ \mathcal {A}_{26}&{:=} {\mathrm {Tr\,}}\mathrm {Sym}\nabla _{I''ca} R_{iabj} \nabla _{J''bd} R_{icdj} \overset{(29), ({\mathrm{A-3}})}{=}\mathcal {T}^{\omega -2}-2\mathcal {M}^{\omega -2}+\mathcal {N}^{\omega -2}.\\ \mathcal {A}_{27}&{:=}{\mathrm {Tr\,}}\mathrm {Sym}\nabla _{I''cb} R_{iabj} \nabla _{J''ad} R_{icdj} \overset{(30)}{=} \mathcal {T}^{\omega -2}-\mathcal {M}^{\omega -2}. \end{aligned}$$
For the remaining terms \(\mathcal {A}_{5}\), \(\mathcal {A}_{19}\) and \(\mathcal {A}_{22}\), the computation is done by induction, by introducing the following sequence \(\mathcal {U}_{\omega -\beta } {:=} -{\mathrm {Tr\,}}{\mathrm {Sym}}\nabla _{I_\beta bc}\nabla _{K_\beta }{\mathrm {Ric}}_{ij}\nabla _{J_\beta }\nabla _{K_\beta } R_{ibcj}\) for \(1\le \beta \le \omega -2\) and \(\mathcal {U}_\omega {:=}\mathcal {A}_5\), where \(K_\beta \), \(I_\beta \) and \(J_\beta \) are multi-indices sets of cardinalities \(\beta \), \(\omega -\beta -2\) and \(\omega -\beta \) respectively. The induction formula is given by
$$\begin{aligned}&\mathcal {U}_{\omega -\beta }=2(\omega -\beta -1)(\omega -\beta -2)\mathcal {U}_{\omega -\beta -1}\\&\quad -2(\omega -\beta -1)^2{\mathrm {Tr\,}}{\mathrm {Sym}}\nabla _{I_\beta bc} \nabla _{K_\beta }{\mathrm {Ric}}_{ij}\nabla _{J_{\beta -2}}\nabla _{K_{\beta }}\Delta R_{ibcj}. \end{aligned}$$
Using (31), we have \({\mathrm {Tr\,}}{\mathrm {Sym}}\nabla _{I_\beta bc}\nabla _{K_\beta }{\mathrm {Ric}}_{ij}\nabla _{J_{\beta -2}}\nabla _{K_{\beta }}\Delta R_{ibcj}=\mathcal {T}^{\omega -\beta -2}-2\mathcal {M}^{\omega -\beta -2}+\mathcal {N}^{\omega -\beta -2}\). By induction on \(\beta \), we prove that
$$\begin{aligned} \mathcal {A}_{5}&=\mathcal {U}_\omega =(\omega -1)!(\omega -2)!\sum _{k=0}^{\omega -2}2^{\omega -k-1} \frac{k+1}{(k!)^2}(\mathcal {T}^{k}-2\mathcal {M}^{k}+ \mathcal {N}^{k})\nonumber \\&\overset{(16),(37)}{=}\sum _{\ell =0}^{[\frac{\omega -2}{2}]} e^{\omega }_\ell (\mathcal {T}_\ell -2\mathcal {M}_\ell + \mathcal {N}_\ell ). \end{aligned}$$
(34)
We have \(\mathcal {A}_{22}{:=} {\mathrm {Tr\,}}\mathrm {Sym}\nabla _{I'c}R_{iabj} \nabla _{J'''abd}R_{icdj}\overset{(30)}{=} -{\mathrm {Tr\,}}\mathrm {Sym}\nabla _{I'c} R_{iabj} \nabla _{J'''abc} R_{ij}\). By contracting an index in \(\mathcal {A}_5\), which corresponds to a covariant derivative of the Riemann tensor and using the last equality, we obtain \(\mathcal {A}_5=2(\omega -1)^2\mathcal {B}_{26}+2(\omega -1)(\omega -2)\mathcal {A}_{22}\). Therefore, by (34), we obtain \(\mathcal {A}_{22} =\sum _{\ell =0}^{[\frac{\omega -3}{2}]}e^{\omega -1}_\ell (\mathcal {T}_\ell -2\mathcal {M}_\ell + \mathcal {N}_\ell )\).
We have \(\mathcal {A}_{19}{:=}{\mathrm {Tr\,}}\mathrm {Sym}\nabla _{I''''abcd} R_{iabj}\nabla _J R_{icdj}=-{\mathrm {Tr\,}}\mathrm {Sym}\nabla _{I''''cd}\Delta R_{ij} \nabla _J R_{icdj}\). By contracting an index in \(\mathcal {A}_5\), which corresponds to a covariant derivative of the Ricci tensor and using the last equality, we obtain \(\mathcal {A}_{5}=2(\omega -1)(\omega -3)\mathcal {A}_{19}+2\omega (\omega -1)\mathcal {A}_{22}\). We deduce that
$$\begin{aligned} \mathcal {A}_{19}= \sum _{\ell =0}^{[\frac{\omega -2}{2}]}\frac{\ell e^{\omega }_\ell }{(\omega -1)(\omega -2)(\omega -3)}(\mathcal {T}_\ell -2\mathcal {M}_\ell + \mathcal {N}_\ell ). \end{aligned}$$
1.2 Combinatorics Formulas
The following identities hold for any nonnegative integer \(\omega \)
$$\begin{aligned} (\omega +2)\sum _{k=\omega }^{2\omega }(k+1) \left( {\begin{array}{c}k\\ \omega \end{array}}\right)&= (\omega +3)^2C(\omega ),\end{aligned}$$
(35)
$$\begin{aligned} (\omega +2)\sum _{k=\omega }^{2\omega -1}(k+1)(2\omega -k)\left( {\begin{array}{c}k\\ \omega \end{array}}\right)&= \omega (\omega +3)C(\omega ), \end{aligned}$$
(36)
where \(C(\omega )=(\omega +1)^2(\omega +2)^2(2\omega +2)![(\omega +3)!]^{-2}\).
Proof
Identities (35) and (36) can be written in a simpler way which can be viewed as special cases (for \(n=2\omega \)) of the following combinatorial identities:
$$\begin{aligned} \sum _{k=\omega }^{n-1}\left( {\begin{array}{c}k\\ \omega \end{array}}\right) =\left( {\begin{array}{c}n\\ \omega +1\end{array}}\right) ,\quad \sum _{k=\omega }^{n-2}(n-k-1)\left( {\begin{array}{c}k\\ \omega \end{array}}\right) =\left( {\begin{array}{c}n\\ \omega +2\end{array}}\right) , \end{aligned}$$
(37)
for all integers \(n\ge \omega \), respectively \(n\ge \omega +2\).
These identities follow by counting in a different way the number of combinations. The first identity is obtained by counting the number of subsets with \((\omega +1)\) elements out of \(n\) elements in the following way: the sets are separated with respect to their largest element. For each \(\omega \le k\le n-1\), \(\left( {\begin{array}{c}k\\ \omega \end{array}}\right) \) counts the subsets of \((\omega +1)\)-elements whose largest element is \(k+1\).
Similarly, the second identity follows by counting \(\left( {\begin{array}{c}n\\ \omega +2\end{array}}\right) \) as follows: the sets are separated with respect to their second largest element. For each \(\omega \le k\le n-2\), \((n-k-1)\left( {\begin{array}{c}k\\ \omega \end{array}}\right) \) is the number of subsets with \(\omega +2\) elements whose second largest element is \(k+1\). Indeed, if the second largest element is \(k+1\), the others \(\omega \) elements of the set which are smaller must form a subset of \(\omega +k\) and the largest element may be any of the remaining \(n-k-1\) elements. \(\square \)
