Inaudibility of sixth order curvature invariants

Abstract

It is known that the spectrum of the Laplace operator on functions of a closed Riemannian manifold does not determine the integrals of the individual fourth order curvature invariants ${{\text {scal}}}^2$, $|{\text {ric}}|^2$, $|R|^2$, which appear as summands in the second heat invariant $a_2$. We study the analogous question for the integrals of the sixth order curvature invariants appearing as summands in $a_3$. Our result is that none of them is determined individually by the spectrum, which can be shown using various examples. In particular, we prove that two isospectral nilmanifolds of Heisenberg type with three-dimensional center are locally isometric if and only if they have the same value of $|\nabla R|^2$. In contrast, any pair of isospectral nilmanifolds of Heisenberg type with centers of dimension $r>3$ does not differ in any curvature invariant of order six, actually not in any curvature invariant of order smaller than 2r. We also prove that this implies that for any $k\in {\mathbb N}$, there exist locally homogeneous manifolds which are not curvature equivalent but do not differ in any curvature invariant of order up to 2k.

Appendix

Proof of Remark 2(iv).  For $A\in {\mathfrak g}$, write $R_A:{\mathfrak g}\times {\mathfrak g}\ni (B,C)\mapsto R(A,B)C\in {\mathfrak g}$, and consider the canonical extension of ${\langle \,\,,\,\rangle }$ to tensors of this form. We start by computing individual formulas for $\langle R_A,R_B\rangle$ because we will need them below in the proof of Lemma 3(ii). For $U,Y\in {\mathfrak v}$ we have, by Lemma 1(ii),

$$\begin{aligned} \bullet&\langle R_U| {{\mathfrak v}\times {\mathfrak v}}\,,R_Y| {{\mathfrak v}\times {\mathfrak v}}\rangle \, =\displaystyle \sum \nolimits _{k,\ell ,a=1}^m\langle R(U,X_k)X_\ell ,X_a\rangle \langle R(Y,X_k)X_\ell ,X_a\rangle \\&\quad = \,\tfrac{1}{16}\sum \nolimits _{k,\ell ,a=1}^m\sum \nolimits _{\beta ,\gamma =1}^r \bigl (\langle j_{Z_\beta }U,X_a\rangle \langle j_{Z_\beta }X_k,X_\ell \rangle -\langle j_{Z_\beta }U,X_\ell \rangle \langle j_{Z_\beta }X_k,X_a\rangle \\&\qquad -2\langle j_{Z_\beta }U,X_k\rangle \langle j_{Z_\beta }X_\ell ,X_a\rangle \bigr )\cdot \\&\cdot \bigl ( \langle j_{Z_\gamma }Y,X_a\rangle \langle j_{Z_\gamma }X_k,X_\ell \rangle -\langle j_{Z_\gamma }Y,X_\ell \rangle \langle j_{Z_\gamma }X_k,X_a\rangle -2\langle j_{Z_\gamma }Y,X_k\rangle \langle j_{Z_\gamma }X_\ell ,X_a\rangle \bigr )\\&\quad = \, \tfrac{1}{16}\sum \nolimits _{\beta ,\gamma =1}^r\bigl ( (1^2+1^2+2^2)\langle j_{Z_\beta }U,j_{Z_\gamma }Y\rangle \langle j_{Z_\beta },j_{Z_\gamma }\rangle \\&\qquad + \, (1-2+1-2-2-2)\langle j_{Z_\beta }U,j_{Z_\gamma }j_{Z_\beta }j_{Z_\gamma }Y\rangle \bigr )\\&\quad = \, \tfrac{3}{8}\sum \nolimits _{\beta ,\gamma =1}^r\bigl (\langle j_{Z_\beta }j_{Z_\gamma }U,Y\rangle {\text {Tr}}( j_{Z_\beta }j_{Z_\gamma })+\langle j_{Z_\beta }j_{Z_\gamma }j_{Z_\beta }j_{Z_\gamma }U,Y\rangle \bigr ),\\ \bullet&\langle R_U| {{\mathfrak v}\times {\mathfrak z}}\,,R_Y| {{\mathfrak v}\times {\mathfrak z}}\rangle =\displaystyle \sum \nolimits _{k=1}^m\sum \nolimits _{\beta ,\gamma =1}^r \langle R(U,X_k)Z_\beta ,Z_\gamma \rangle \langle R(Y,X_k)Z_\beta ,Z_\gamma \rangle \\&\quad =\tfrac{1}{16}\sum \nolimits _{k=1}^m\sum \nolimits _{\beta ,\gamma =1}^r \langle [j_{Z_\beta },j_{Z_\gamma }]U,X_k\rangle \langle [j_{Z_\beta },j_{Z_\gamma }]Y,X_k\rangle \\&\quad = -\tfrac{1}{16}\sum \nolimits _{\beta ,\gamma =1}^r \langle [j_{Z_\beta },j_{Z_\gamma }]^2 U,Y\rangle \\&\quad =-\tfrac{1}{8}\sum \nolimits _{\beta ,\gamma =1}^r\langle j_{Z_\beta }j_{Z_\gamma }j_{Z_\beta }j_{Z_\gamma }U,Y\rangle +\tfrac{1}{8}\sum _{\beta =1}^r \langle j_{Z_\beta }Jj_{Z_\beta }U,Y\rangle ,\\ \bullet&\displaystyle \langle R_U| {{\mathfrak z}\times {\mathfrak v}}\,,R_Y| {{\mathfrak z}\times {\mathfrak v}}\rangle +\langle R_U| {{\mathfrak z}\times {\mathfrak z}}\,,R_Y| {{\mathfrak z}\times {\mathfrak z}}\rangle \\&\quad =2\sum \nolimits _{k=1}^m\sum _{\beta ,\gamma =1}^r\langle R(U,Z_\beta )Z_\gamma ,X_k\rangle \langle R(Y,Z_\beta )Z_\gamma ,X_k\rangle \quad \quad \;\\&\quad =\tfrac{1}{8}\sum \nolimits _{k=1}^m\sum _{\beta ,\gamma =1}^r\langle j_{Z_\gamma }U,j_{Z_\beta }X_k\rangle \langle j_{Z_\gamma }Y,j_{Z_\beta }X_k\rangle \\&\quad =\tfrac{1}{8}\sum \nolimits _{\beta ,\gamma =1}^r\langle j_{Z_\beta }j_{Z_\gamma }U,j_{Z_\beta }j_{Z_\gamma }Y\rangle \\&\quad =\tfrac{1}{8}\sum \nolimits _{\beta =1}^r\langle j_{Z_\beta }Jj_{Z_\beta }U,Y\rangle . \end{aligned}$$

Hence,

$$\begin{aligned} \langle R_U,R_Y\rangle= & {} \sum \nolimits _{\beta ,\gamma =1}^r\bigl (\tfrac{3}{8}\langle j_{Z_\beta }j_{Z_\gamma }U,Y\rangle {\text {Tr}}(j_{Z_\beta }j_{Z_\gamma }) +\tfrac{1}{4}\langle j_{Z_\beta }j_{Z_\gamma }j_{Z_\beta }j_{Z_\gamma }U,Y\rangle \bigr ) \nonumber \\&\quad +\tfrac{1}{4}\sum \nolimits _{\beta =1}^r\langle j_{Z_\beta }Jj_{Z_\beta }U,Y\rangle . \end{aligned}$$

(16)

For $W\in {\mathfrak z}$ we have $R_W| {{\mathfrak z}\times {\mathfrak z}}=0$ and

$$\begin{aligned} \bullet&|R_W| {{\mathfrak v}\times {\mathfrak v}}|^2+|R_W| {{\mathfrak v}\times {\mathfrak z}}|^2 =2\displaystyle \sum _{k,\ell =1}^m\sum \nolimits _{\alpha =1}^r\langle R(W,X_k)X_\ell ,Z_\alpha \rangle ^2\\&\quad = \, \tfrac{1}{8}\sum \nolimits _{k,\ell =1}^m\sum \nolimits _{\alpha =1}^r\langle j_W X_\ell ,j_{Z_\alpha }X_k\rangle ^2=\tfrac{1}{8}\sum _{\alpha =1}^2|j_Wj_{Z_\alpha }|^2=\tfrac{1}{8}{\text {Tr}}(Jj_W^2),\\ \bullet&\displaystyle |R_W| {{\mathfrak z}\times {\mathfrak v}}|^2=\displaystyle \sum \nolimits _{k,\ell =1}^m\sum \nolimits _{\alpha =1}^r\langle R(W,Z_\alpha )X_k,X_\ell \rangle ^2 \\&\quad = \, \tfrac{1}{16}\sum \nolimits _{k,\ell =1}^m\sum \nolimits _{\alpha =1}^r\langle [j_W,j_{Z_\alpha }]X_k,X_\ell \rangle ^2 =\tfrac{1}{16}\sum \nolimits _{\alpha =1}^r|j_Wj_{Z_\alpha }-j_{Z_\alpha }j_W|^2 \\&\quad = \, \tfrac{1}{8}{\text {Tr}}(Jj_W^2)-\tfrac{1}{8}\sum \nolimits _{\alpha =1}^r {\text {Tr}}(j_Wj_{Z_\alpha }j_Wj_{Z_\alpha }) \end{aligned}$$

and thus for $Z,W\in {\mathfrak z}$, using polarization,

$$\begin{aligned} \langle R_Z,R_W\rangle =\tfrac{1}{8}{\text {Tr}}(J(j_Zj_W+j_Wj_Z))-\tfrac{1}{8}\sum \nolimits _{\alpha =1}^r{\text {Tr}}(j_Zj_{Z_\alpha }j_Wj_{Z_\alpha }). \end{aligned}$$

(17)

Moreover, $\langle R_X,R_Z\rangle =0$ for all $X\in {\mathfrak v}$, $Z\in {\mathfrak z}$ by Lemma 1(ii). Using (16) and (17), we obtain

$$\begin{aligned} |R|^2= & {} \sum \nolimits _{k=1}^m\langle R_{X_k},R_{X_k}\rangle +\sum \nolimits _{\alpha =1}^r\langle R_{Z_\alpha },R_{Z_\alpha }\rangle \\= & {} \tfrac{3}{8}I_{\alpha \beta |\alpha \beta }+\tfrac{1}{4}I_{\alpha \beta \alpha \beta }+\tfrac{1}{4}I_{\alpha \alpha \beta \beta } +\tfrac{1}{4}I_{\alpha \alpha \beta \beta }-\tfrac{1}{8}I_{\alpha \beta \alpha \beta }, \end{aligned}$$

from which the statement follows. $\square$

Proof of Lemma 3.

• (i) First note that by Lemma 1 and since J is symmetric,

  $$\begin{aligned}&\sum \nolimits _{k,\ell ,a,b=1}^m{\text {ric}}(X_k,X_\ell ){\text {ric}}(X_a,X_b)\langle R(X_k,X_a)X_\ell ,X_b\rangle \\&\quad =\tfrac{1}{4}\sum \nolimits _{k,\ell ,a,b=1}^m\langle JX_k,X_\ell \rangle \langle JX_a,X_b\rangle \cdot \\&\cdot \sum \nolimits _{\beta =1}^r \bigl (\tfrac{1}{4}\langle j_{Z_\beta }X_a,X_\ell \rangle \langle j_{Z_\beta }X_k,X_b\rangle -\tfrac{1}{4} \langle j_{Z_\beta }X_k,X_\ell \rangle \langle j_{Z_\beta }X_a,X_b\rangle \\&\qquad -\tfrac{1}{2}\langle j_{Z_\beta }X_k,X_a\rangle \langle j_{Z_\beta }X_\ell ,X_b\rangle \bigr )\\&\quad = \tfrac{1}{16}\sum \nolimits _{k,a=1}^m\sum \nolimits _{\beta =1}^r\bigl (\langle j_{Z_\beta }X_a,JX_k\rangle \langle j_{Z_\beta }X_k,JX_a\rangle -\langle j_{Z_\beta }X_k,JX_k\rangle \langle j_{Z_\beta }X_a,JX_a\rangle \\&\qquad -2\langle j_{Z_\beta }X_k,X_a\rangle \langle j_{Z_\beta }JX_k,JX_a\rangle \bigr )\\&\quad =\tfrac{1}{16}\sum \nolimits _{\beta =1}^r(\langle -j_{Z_\beta }J,Jj_{Z_\beta }\rangle -0-2\langle j_{Z_\beta },Jj_{Z_\beta }J\rangle ) \\&\quad =\tfrac{1}{16}\sum \nolimits _{\beta =1}^r3{\text {Tr}}(Jj_{Z_\beta }Jj_{Z_\beta }) =\tfrac{3}{16}I_{\alpha \alpha \beta \gamma \gamma \beta }. \end{aligned}$$

  Moreover, ${\text {ric}}({\mathfrak v},{\mathfrak z})=0$, $\langle R({\mathfrak z},{\mathfrak z}){\mathfrak z},{\mathfrak z}\rangle =0$, and

  $$\begin{aligned}&2\cdot \sum \nolimits _{k,\ell =1}^m\sum \nolimits _{\beta ,\gamma =1}^r{\text {ric}}(X_k,X_\ell ){\text {ric}}(Z_\beta ,Z_\gamma ) \langle R(X_k,Z_\beta )X_\ell ,Z_\gamma \rangle \\&\quad =\tfrac{1}{16}\sum \nolimits _{k,\ell =1}^m\sum \nolimits _{\beta ,\gamma =1}^r\langle JX_k,X_\ell \rangle {\text {Tr}}(j_{Z_\beta }j_{Z_\gamma }) \langle j_{Z_\beta }j_{Z_\gamma }X_k,X_\ell \rangle \\&\quad =\tfrac{1}{16}\sum \nolimits _{\beta ,\gamma =1}^r {\text {Tr}}(Jj_{Z_\beta }j_{Z_\gamma }){\text {Tr}}(j_{Z_\beta }j_{Z_\gamma })\\&\quad =\tfrac{1}{16}I_{\alpha \alpha \beta \gamma |\beta \gamma }. \end{aligned}$$

  Thus,

  $$\begin{aligned} (*)=\tfrac{3}{16}I_{\alpha \alpha \beta \gamma \gamma \beta }+\tfrac{1}{16}I_{\alpha \alpha \beta \gamma |\beta \gamma }. \end{aligned}$$

• (ii) By definition of $(**)$ and by Lemma 1(iii),

  $$\begin{aligned} (**)=\sum \nolimits _{k,\ell =1}^m\tfrac{1}{2}\langle JX_k,X_\ell \rangle \langle R_{X_k},R_{X_\ell }\rangle -\sum \nolimits _{\alpha ,\beta =1}^r\tfrac{1}{4}{\text {Tr}}(j_{Z_\alpha }j_{Z_\beta })\langle R_{Z_\alpha },R_{Z_\beta }\rangle . \end{aligned}$$

  Thus, using (16) and (17),

  $$\begin{aligned}&(**)= \tfrac{1}{2}\sum \nolimits _{k=1}^m\Bigl (\sum \nolimits _{\beta ,\gamma =1}^r \bigl (\tfrac{3}{8}\langle j_{Z_\beta }j_{Z_\gamma }X_k,JX_k\rangle {\text {Tr}}(j_{Z_\beta }j_{Z_\gamma })\\&\quad +\tfrac{1}{4}\langle j_{Z_\beta }j_{Z_\gamma }j_{Z_\beta }j_{Z_\gamma }X_k,JX_k\rangle \bigr )\\&\quad +\tfrac{1}{4}\sum \nolimits _{\beta =1}^r\langle j_{Z_\beta }Jj_{Z_\beta }X_k,JX_k\rangle \Bigr )\\&\quad -\tfrac{1}{4}\sum \nolimits _{\beta ,\gamma =1}^r{\text {Tr}}(j_{Z_\beta }j_{Z_\gamma })\bigl (\tfrac{1}{8}{\text {Tr}}(J(j_{Z_\beta }j_{Z_\gamma } +j_{Z_\gamma }j_{Z_\beta }))\\&\quad -\tfrac{1}{8}\sum \nolimits _{\alpha =1}^r{\text {Tr}}(j_{Z_\beta }j_{Z_\alpha }j_{Z_\gamma }j_{Z_\alpha })\bigr )\\&\qquad \,\,= \tfrac{3}{16}I_{\alpha \alpha \beta \gamma |\beta \gamma }+\tfrac{1}{8}I_{\alpha \alpha \beta \gamma \beta \gamma } +\tfrac{1}{8}I_{\alpha \alpha \beta \gamma \gamma \beta } -\tfrac{1}{16}I_{\alpha \alpha \beta \gamma |\beta \gamma }+\tfrac{1}{32}I_{\beta \alpha \gamma \alpha |\beta \gamma }\\&\qquad \,\,= \tfrac{1}{8}I_{\alpha \alpha \beta \gamma |\beta \gamma }+\tfrac{1}{8}I_{\alpha \alpha \beta \gamma \beta \gamma } +\tfrac{1}{8}I_{\alpha \alpha \beta \gamma \gamma \beta }+\tfrac{1}{32}I_{\alpha \gamma \beta \gamma |\alpha \beta }. \end{aligned}$$

• (iii) For $X,Y\in {\mathfrak v}$ and $Z\in {\mathfrak z}$, we have $(\nabla _Y{\text {ric}})| {{\mathfrak v}\times {\mathfrak v}}=0$, $(\nabla _Y{\text {ric}})| {{\mathfrak z}\times {\mathfrak z}}=0$ and

  $$\begin{aligned} ((\nabla _Y{\text {ric}})(Z,X))^2= & {} (\tfrac{1}{2}{\text {ric}}(X,j_ZY)-\tfrac{1}{2}{\text {ric}}([Y,X],Z))^2\\= & {} (\tfrac{1}{4}\langle JX,j_ZY\rangle +\tfrac{1}{8}\sum \nolimits _{\beta =1}^r{\text {Tr}}(j_{Z_\beta }j_Z)\langle j_{Z_\beta }Y,X\rangle )^2,\text { hence} \end{aligned}$$
  $$\begin{aligned} |(\nabla _Y{\text {ric}})|^2= & {} 2\sum \nolimits _{\ell =1}^m\sum \nolimits _{\gamma =1}^r\bigl (\tfrac{1}{16} \sum \nolimits _{\beta =1}^r\langle JX_\ell ,j_{Z_\gamma }Y\rangle {\text {Tr}}(j_{Z_\beta }j_{Z_\gamma })\langle j_{Z_\beta }Y,X_\ell \rangle \\&\quad +\tfrac{1}{16}\langle JX_\ell ,j_{Z_\gamma }Y\rangle ^2\\&\quad +\tfrac{1}{64}\sum \nolimits _{\alpha ,\beta =1}^r{\text {Tr}}(j_{Z_\alpha }j_{Z_\gamma }){\text {Tr}}(j_{Z_\beta }j_{Z_\gamma }) \langle j_{Z_\alpha }Y,X_\ell \rangle \langle j_{Z_\beta }Y,X_\ell \rangle \bigr )\\= & {} \sum \nolimits _{\gamma =1}^r\bigl (\tfrac{1}{8}|Jj_{Z_\gamma }Y|^2 +\tfrac{1}{8}\sum _{\beta =1}^r{\text {Tr}}(j_{Z_\beta }j_{Z_\gamma })\langle j_{Z_\beta }Y,Jj_{Z_\gamma }Y\rangle \\&\quad +\tfrac{1}{32}\sum \nolimits _{\alpha ,\beta =1}^r{\text {Tr}}(j_{Z_\alpha }j_{Z_\gamma }){\text {Tr}}(j_{Z_\beta }j_{Z_\gamma }) \langle j_{Z_\alpha }Y,j_{Z_\beta }Y\rangle \bigr ).\text { Thus,}\\ |(\nabla {\text {ric}})| {{\mathfrak v}}|^2= & {} \sum \nolimits _{\gamma =1}^r\bigl (\tfrac{1}{8}|Jj_{Z_\gamma }|^2 -\tfrac{1}{8}\sum \nolimits _{\beta =1}^r{\text {Tr}}(j_{Z_\beta }j_{Z_\gamma }){\text {Tr}}(Jj_{Z_\gamma }j_{Z_\beta })\\&\quad -\tfrac{1}{32} \sum \nolimits _{\alpha ,\beta =1}^r{\text {Tr}}(j_{Z_\alpha }j_{Z_\gamma }){\text {Tr}}(j_{Z_\beta }j_{Z_\gamma }) {\text {Tr}}(j_{Z_\alpha }j_{Z_\beta })\bigr )\\= & {} -\tfrac{1}{8}{\text {Tr}}(J^3)-\tfrac{1}{8}I_{\alpha \alpha \beta \gamma |\beta \gamma } -\tfrac{1}{32}I_{\alpha \beta |\alpha \gamma |\beta \gamma }, \end{aligned}$$

  where $|(\nabla {\text {ric}})| {{\mathfrak v}}|^2$ denotes $\sum \nolimits _{k=1}^m|\nabla _{X_k}{\text {ric}}|^2$. For $X,Y\in {\mathfrak v}$ and $W\in {\mathfrak z}$, we have $(\nabla _W{\text {ric}})| {{\mathfrak v}\times {\mathfrak z}}=0$, $(\nabla _W{\text {ric}})| {{\mathfrak z}\times {\mathfrak v}}=0$, $(\nabla _W{\text {ric}})| {{\mathfrak z}\times {\mathfrak z}}=0$, and

  $$\begin{aligned} ((\nabla _W{\text {ric}})(X,Y))^2= & {} (-\tfrac{1}{2}{\text {ric}}(j_WX,Y)-\tfrac{1}{2}{\text {ric}}(X,j_WY))^2 \\= & {} (\tfrac{1}{4}\langle Jj_WX,Y\rangle +\tfrac{1}{4}\langle JX,j_WY\rangle )^2, \end{aligned}$$

  hence

  $$\begin{aligned} |\nabla _W{\text {ric}}|^2= & {} \tfrac{1}{16}\sum _{k,\ell =1}^m \bigl (\langle Jj_WX_k,X_\ell \rangle ^2+\langle JX_k,j_WX_\ell \rangle ^2+2\langle Jj_WX_k,X_\ell \rangle \langle JX_k,j_WX_\ell \rangle \bigr )\\= & {} \tfrac{1}{8}|Jj_W|^2-\tfrac{1}{8}\langle Jj_W,j_WJ\rangle .\text { Thus,}\\ |(\nabla {\text {ric}})| {{\mathfrak z}}|^2= & {} \sum _{\beta =1}^r \bigl (\tfrac{1}{8}|Jj_{Z_\beta }|^2+\tfrac{1}{8}{\text {Tr}}(Jj_{Z_\beta }Jj_{Z_\beta })\bigr ) =-\tfrac{1}{8}{\text {Tr}}(J^3)+\tfrac{1}{8}I_{\alpha \alpha \beta \gamma \gamma \beta }, \end{aligned}$$

  where $|(\nabla {\text {ric}})| {{\mathfrak z}}|^2$ denotes $\sum _{\alpha =1}^r|\nabla _{Z_\alpha }{\text {ric}}|^2$. So,

  $$\begin{aligned} |\nabla {\text {ric}}|^2= & {} |(\nabla {\text {ric}})| {{\mathfrak v}}|^2+|(\nabla {\text {ric}})| {{\mathfrak z}}|^2\\= & {} -\tfrac{1}{4}{\text {Tr}}(J^3)-\tfrac{1}{8}I_{\alpha \alpha \beta \gamma |\beta \gamma } -\tfrac{1}{32}I_{\alpha \beta |\alpha \gamma |\beta \gamma }+\tfrac{1}{8}I_{\alpha \alpha \beta \gamma \gamma \beta }. \end{aligned}$$

$\square$

Proof of Lemma 4

• (i) The various contributions

  $$\begin{aligned} \sum \langle (\nabla _AR)(B,C)D,E\rangle ^2 \end{aligned}$$

  (18)

  to $|\nabla R|^2$, where each of A, B, C, D, E runs through either the orthonormal basis $\{X_1,\ldots ,X_m\}$ of ${\mathfrak v}$ or the orthonormal basis $\{Z_1,\ldots ,Z_r\}$ of ${\mathfrak z}$, can by Remark 7 be nonzero only in the cases where ${\mathfrak v}$ occurs an even number of times. If ${\mathfrak v}$ occurs exactly twice then, again by Remark 7, each $\langle (\nabla _AR)(B,C)D,E\rangle$ is a linear combination of terms of the type $\langle j_{Z_{\alpha _1}}j_{Z_{\alpha _2}}j_{Z_{\alpha _3}}X_{\ell _1}X_{\ell _2}\rangle$, where $(X_{\ell _1},X_{\ell _2},Z_{\alpha _1},Z_{\alpha _2},Z_{\alpha _3})$ is just some permutation of (A, B, C, D, E). The sum in (18) will thus be a linear combination of sums of the type

  $$\begin{aligned} \sum \langle j_{Z_{\alpha _{s_1}}}j_{Z_{\alpha _{s_2}}}j_{Z_{\alpha _{s_3}}}X_{\ell _{u_1}},X_{\ell _{u_2}}\rangle \langle j_{Z_{\alpha _{s_4}}}j_{Z_{\alpha _{s_5}}}j_{Z_{\alpha _{s_6}}}X_{\ell _{u_1}},X_{\ell _{u_2}}\rangle , \end{aligned}$$

  where $(s_1,s_2,s_3)$ and $(s_4,s_5,s_6)$ are the same up to permutation, and the summation is done w.r.t. pairs of equal indices $s_i$ and $u_j$. Summation over equal pairs of $u_j$ yields

  $$\begin{aligned} -\sum {\text {Tr}}(j_{Z_{\alpha _{s_6}}}j_{Z_{\alpha _{s_5}}}j_{Z_{\alpha _{s_4}}}j_{Z_{\alpha _{s_1}}} j_{Z_{\alpha _{s_2}}}j_{Z_{\alpha _{s_3}}}), \end{aligned}$$

  which equals $-I_{s_6s_5s_4s_1s_2s_3}$, one of our invariants from Definition 3 in which only subtuples of even length occur (in this case, only one subtuple, and this one of length six). Hence, sums as in (18) with exactly two occurrences of ${\mathfrak v}$ contribute only to the term $L_1$ from the assertion. Therefore it remains to consider sums as in (18) with exactly four occurrences of ${\mathfrak v}$. Due to the symmetries of R, the contribution of such sums is equal to

  $$\begin{aligned} \sum \langle (\nabla _WR)(X,Y)U,V\rangle ^2 +4\sum \langle (\nabla _XR)(W,Y)U,V\rangle ^2, \end{aligned}$$

  (19)

  where both sums are taken over $X,Y,U,V\in \{X_1,\ldots ,X_m\}$, $W\in \{Z_1,\ldots ,Z_r\}$. For the first term in (19), we note using Lemma 1(i), (ii) and the skew-symmetry of the maps $j_W$, $j_{Z_\alpha }$ that $\langle (\nabla _WR)(X,Y)U,V\rangle$ is the sum of the following twelve summands:

  • (a) $-\langle \nabla _W\nabla _X\nabla _YU,V\rangle =-\frac{1}{8}\sum _{\beta }\langle j_Wj_{Z_\beta }X,V\rangle \langle j_{Z_\beta }Y,U\rangle$,

  • (b) $\langle \nabla _W\nabla _Y\nabla _XU,V\rangle =\frac{1}{8}\sum _\beta \langle j_Wj_{Z_\beta }Y,V\rangle \langle j_{Z_\beta }X,U\rangle$,

  • (c) $\langle \nabla _W\nabla _{[X,Y]}U,V\rangle =\frac{1}{4}\sum _\beta \langle j_Wj_{Z_\beta }U,V\rangle \langle j_{Z_\beta }X,Y\rangle$,

  • (d) $\langle \nabla _{\nabla _WX}\nabla _YU,V\rangle =\frac{1}{8}\sum _\beta \langle j_{Z_\beta }j_WX,V\rangle \langle j_{Z_\beta }Y,U\rangle$,

  • (e) $-\langle \nabla _Y\nabla _{\nabla _WX}U,V\rangle =-\frac{1}{8}\sum _\beta \langle j_{Z_\beta }j_WX,U\rangle \langle j_{Z_\beta }Y,V\rangle$,

  • (f) $-\langle \nabla _{[\nabla _WX,Y]}U,V\rangle =-\frac{1}{4}\sum _\beta \langle j_{Z_\beta }j_WX,Y\rangle \langle j_{Z_\beta }U,V\rangle$,

  • (g) $\langle \nabla _X\nabla _{\nabla _WY}U,V\rangle =\frac{1}{8}\sum _\beta \langle j_{Z_\beta }j_WY,U\rangle \langle j_{Z_\beta }X,V\rangle$,

  • (h) $-\langle \nabla _{\nabla _WY}\nabla _XU,V\rangle =-\frac{1}{8}\sum _\beta \langle j_{Z_\beta }j_WY,V\rangle \langle j_{Z_\beta }X,U\rangle$,

  • (i) $-\langle \nabla _{[X,\nabla _WY]}U,V\rangle =\frac{1}{4}\sum _\beta \langle j_Wj_{Z_\beta }X,Y\rangle \langle j_{Z_\beta }U,V\rangle$,

  • (j) $\langle \nabla _X\nabla _Y\nabla _WU,V\rangle =-\frac{1}{8}\sum _\beta \langle j_Wj_{Z_\beta }Y,U\rangle \langle j_{Z_\beta }X,V\rangle$,

  • (k) $-\langle \nabla _Y\nabla _X\nabla _WU,V\rangle =\frac{1}{8}\sum _\beta \langle j_Wj_{Z_\beta }X,U\rangle \langle j_{Z_\beta }Y,V\rangle$,

  • (l) $-\langle \nabla _{[X,Y]}\nabla _WU,V\rangle =-\frac{1}{4}\sum _\beta \langle j_{Z_\beta }j_WU,V\rangle \langle j_{Z_\beta }X,Y\rangle$,

  where the sums are taken over $\beta \in \{1,\ldots ,r\}$. Now $\langle (\nabla _WR)(X,Y)U,V\rangle ^2$ is the square of the sum of the twelve terms. The square of each single one of them will just lead to a contribution to $L_1$: For example, the square of the term in (a) is $\frac{1}{64}$ times

  $$\begin{aligned} \sum \nolimits _{\beta ,\gamma } \langle j_Wj_{Z_\beta }X,V\rangle \langle j_{Z_\beta }Y,U\rangle \langle j_Wj_{Z_\gamma }X,V\rangle \langle j_{Z_\gamma }Y,U\rangle , \end{aligned}$$

  which after summation over $X,Y,U,V\in \{X_1,\ldots ,X_m\}$ gives

  $$\begin{aligned} -\sum \nolimits _{\beta ,\gamma }{\text {Tr}}(j_Wj_Wj_{Z_\beta }j_{Z_\gamma }){\text {Tr}}(j_{Z_\beta }j_{Z_\gamma }); \end{aligned}$$

  summation over $W\in \{Z_1,\ldots ,Z_r\}$ thus yields $-I_{\alpha \alpha \beta \gamma |\beta \gamma }$, another invariant in which only subtuples of even lengths (here, four and two) occur. Next, consider the product of the terms in (a) and (b) which is $-\frac{1}{64}$ times

  $$\begin{aligned} \sum \nolimits _{\beta ,\gamma } \langle j_Wj_{Z_\beta }X,V\rangle \langle j_{Z_\beta }Y,U\rangle \langle j_Wj_{Z_\gamma }Y,V\rangle \langle j_{Z_\gamma }X,U\rangle . \end{aligned}$$

  One easily checks that this leads to an invariant with just one subtuple of length six. The technical reason is that here, there is no way to group the four factors into subsets which would not be linked to each other by the occurrence of any common vectors from $\{X,Y,U,V\}$. The only pairings of different terms from (a)–(l) above where this does not happen are the following twelve:

  $$\begin{aligned} ((a)\text { or }(d))\longleftrightarrow ((g)\text { or }(j)), \nonumber \\ ((b)\text { or }(h))\longleftrightarrow ((e)\text { or }(k)), \nonumber \\ ((c)\text { or }(l))\longleftrightarrow ((f)\text { or }(i)), \end{aligned}$$

  (20)

  For example, the product of the terms in (a) and (g) is

  $$\begin{aligned} -\tfrac{1}{64}\sum \nolimits _{\beta ,\gamma } \langle j_Wj_{Z_\beta }X,V\rangle \langle j_{Z_\beta }Y,U\rangle \langle j_{Z_\gamma }j_WY,U\rangle \langle j_{Z_\gamma }X,V\rangle \end{aligned}$$

  (21)

  which after summation over X, Y, U, V becomes

  $$\begin{aligned} -\frac{1}{64}\sum \nolimits _{\beta ,\gamma }{\text {Tr}}(j_{Z_\gamma }j_Wj_{Z_\beta }){\text {Tr}}(j_{Z_\beta }j_{Z_\gamma }j_W). \end{aligned}$$

  Summation over W finally yields $-\tfrac{1}{64}I_{\gamma \alpha \beta |\beta \gamma \alpha } =-\tfrac{1}{64}I_{\alpha \beta \gamma |\alpha \beta \gamma }$. Similarly, the product of the terms in (a) and (j) is

  $$\begin{aligned} \tfrac{1}{64}\sum \nolimits _{\beta ,\gamma } \langle j_Wj_{Z_\beta }X,V\rangle \langle j_{Z_\beta }Y,U\rangle \langle j_Wj_{Z_\gamma }Y,U\rangle \langle j_{Z_\gamma }X,V\rangle \end{aligned}$$

  which gives $-\frac{1}{64}{\text {Tr}}(j_{Z_\gamma } j_W j_{Z_\beta }){\text {Tr}}(j_{Z_\gamma } j_W j_{Z_\beta }) =-\frac{1}{64}I_{\alpha \beta \gamma |\alpha \beta \gamma }$ again. For each of the pairings from (20), note that whenever the two terms to be paired differ in sign, they also differ in the order of $j_W$ and $j_{Z_\beta }$ in their first factors. Just as we saw for $(a)\leftrightarrow (g)$ and $(a)\leftrightarrow (j)$, this leads each time to a negative multiple of $I_{\alpha \beta \gamma |\alpha \beta \gamma }$. Altogether, we obtain

  $$\begin{aligned} 2\cdot \bigl (4\cdot (-\tfrac{1}{64})+4\cdot (-\tfrac{1}{64})+4\cdot (-\tfrac{1}{16})\bigr )I_{\alpha \beta \gamma |\alpha \beta \gamma } =-\tfrac{3}{4} I_{\alpha \beta \gamma |\alpha \beta \gamma } \end{aligned}$$

  as the contribution to $|\nabla R|^2$ of the first summand in (19), apart from its contributions to $L_1$. For the second summand in (19), we compute that $\langle (\nabla _X R)(W,Y)U,V\rangle$ is the sum of

  • (a’) $-\langle \nabla _X\nabla _W\nabla _YU,V\rangle =0$,

  • (b’) $\langle \nabla _X\nabla _Y\nabla _WU,V\rangle =-\frac{1}{8}\sum _\beta \langle j_Wj_{Z_\beta }Y,U\rangle \langle j_{Z_\beta }X,V\rangle$,

  • (c’) $\langle \nabla _X\nabla _{[W,Y]}U,V\rangle =0$,

  • (d’) $\langle \nabla _{\nabla _XW}\nabla _YU,V\rangle =\frac{1}{8}\sum _\beta \langle j_{Z_\beta }j_WX,V\rangle \langle j_{Z_\beta }Y,U\rangle$,

  • (e’) $-\langle \nabla _Y\nabla _{\nabla _XW}U,V\rangle =-\frac{1}{8}\sum _\beta \langle j_{Z_\beta }j_WX,U\rangle \langle j_{Z_\beta }Y,V\rangle$,

  • (f’) $-\langle \nabla _{[\nabla _XW,Y]}U,V\rangle =-\frac{1}{4}\sum _\beta \langle j_{Z_\beta }j_WX,Y\rangle \langle j_{Z_\beta }U,V\rangle$,

  • (g’) $\langle \nabla _W\nabla _{\nabla _XY}U,V\rangle =\frac{1}{8}\sum _\beta \langle j_Wj_{Z_\beta }U,V\rangle \langle j_{Z_\beta }X,Y\rangle$,

  • (h’) $-\langle \nabla _{\nabla _XY}\nabla _WU,V\rangle =-\frac{1}{8}\sum _\beta \langle j_{Z_\beta }j_WU,V\rangle \langle j_{Z_\beta }X,Y\rangle$,

  • (i’) $-\langle \nabla _{[W,\nabla _XY]}U,V\rangle =0$,

  • (j’) $\langle \nabla _W\nabla _Y\nabla _XU,V\rangle =\frac{1}{8}\sum _\beta \langle j_Wj_{Z_\beta }Y,V\rangle \langle j_{Z_\beta }X,U\rangle$,

  • (k’) $-\langle \nabla _Y\nabla _W\nabla _XU,V\rangle =0$,

  • (l’) $-\langle \nabla _{[W,Y]}\nabla _XU,V\rangle =0$.

  Similarly as above, the only pairings which do not just contribute to $L_1$ now are

  $$\begin{aligned} (b') \longleftrightarrow (d'),\quad (e') \longleftrightarrow (j'),\quad (f') \longleftrightarrow ((g')\text { or }(h')). \end{aligned}$$

  Again, each of these pairings gives a negative multiple of $I_{\alpha \beta \gamma |\alpha \beta \gamma }$. All in all, we obtain

  $$\begin{aligned} 4\cdot 2\cdot \bigl (-\tfrac{1}{64}-\tfrac{1}{64}+2\cdot (-\tfrac{1}{32})\bigr )I_{\alpha \beta \gamma |\alpha \beta \gamma } =-\tfrac{3}{4} I_{\alpha \beta \gamma |\alpha \beta \gamma } \end{aligned}$$

  as the contribution to $|\nabla R|^2$ of the second summand in (19), apart from its contributions to $L_1$. The statement now follows by $-\frac{3}{4}-\frac{3}{4}=-\frac{3}{2}$.

• (ii) The various contributions

  $$\begin{aligned} \sum \langle R(A,B)C,D\rangle \langle R(C,D)E,F\rangle \langle R(E,F)A,B\rangle \end{aligned}$$

  (22)

  to $\hat{R}$, where each of A, B, C, D, E, F runs through either the orthonormal basis $\{X_1,\ldots , X_m\}$ of ${\mathfrak v}$ or the orthonormal basis $\{Z_1,\ldots ,Z_r\}$ of ${\mathfrak z}$, can by Lemma 1(ii) be nonzero only in the cases where each of the tuples

  $$\begin{aligned} (A,B,C,D),\quad (C,D,E,F), \quad (E,F,A,B) \end{aligned}$$

  (23)

  contains either two or four vectors from ${\mathfrak v}$. If each of them contains exactly two vectors from ${\mathfrak v}$, then each summand in (22) is, again by Lemma 1(ii), a linear combination of products of three terms of the form $\langle j_{Z_{\alpha _1}}j_{Z_{\alpha _2}}X_{\ell _1},X_{\ell _2}\rangle$. The sum in (22) will thus be a linear combination of sums of the type

  $$\begin{aligned} \sum \langle j_{Z_{\alpha _{s_1}}}j_{Z_{\alpha _{s_2}}}X_{\ell _{u_1}},X_{\ell _{u_2}}\rangle \langle j_{Z_{\alpha _{s_3}}}j_{Z_{\alpha _{s_4}}}X_{\ell _{u_3}},X_{\ell _{u_4}}\rangle \langle j_{Z_{\alpha _{s_5}}}j_{Z_{\alpha _{s_6}}}X_{\ell _{u_5}},X_{\ell _{u_6}}\rangle , \end{aligned}$$

  with each $s_i$ and each $u_j$ occurring exactly twice. Here, summation over equal pairs of $u_j$ will obviously always lead to invariants $I_{k_1\ldots k_\lambda |\ldots |k_\mu \ldots k_6}$ in which all subtuples are of even length (6, or 4 and 2, or three times 2). Hence, such sums will contribute only to the term $L_2$ from the assertion. So let at least one of the tuples from (23) consists of vectors in ${\mathfrak v}$. If $A,B,C,D\in {\mathfrak v}$ then either E, F must both be in ${\mathfrak v}$ or both in ${\mathfrak z}$. Therefore, the contributions of sums as in (22) where at least one of the tuples from (23) consists of vectors in ${\mathfrak v}$ is equal to

  $$\begin{aligned}&\sum \langle R(X,Y)U,V\rangle \langle R(U,V)S,T\rangle \langle R(S,T)X,Y\rangle \nonumber \\&\quad + \, 3\sum \langle R(X,Y)U,V\rangle \langle R(U,V)Z,W\rangle \langle R(Z,W)X,Y\rangle , \end{aligned}$$

  (24)

  where the first sum is taken over $X,Y,U,V,S,T\in \{X_1,\ldots ,X_m\}$ and the second sum over $X,Y,U,V\in \{X_1,\ldots ,X_m\}$ and $Z,W\in \{Z_1,\ldots ,Z_r\}$. For the first term in (24), we note that

  $$\begin{aligned}&\langle R(X,Y)U,V \rangle \langle R(U,V)S,T\rangle \langle R(S,T)X,Y\rangle = \nonumber \\&\sum \nolimits _{\alpha ,\beta ,\gamma =1}^r \bigl (\tfrac{1}{4}\langle j_{Z_\alpha }Y,U\rangle \langle j_{Z_\alpha }X,V\rangle -\tfrac{1}{4}\langle j_{Z_\alpha }X,U\rangle \langle j_{Z_\alpha }Y,V\rangle -\tfrac{1}{2}\langle j_{Z_\alpha }X,Y\rangle \langle j_{Z_\alpha }U,V\rangle \bigr ) \nonumber \\&\quad \cdot \bigl (\tfrac{1}{4}\langle j_{Z_\beta }V,S\rangle \langle j_{Z_\beta }U,T\rangle -\tfrac{1}{4}\langle j_{Z_\beta }U,S\rangle \langle j_{Z_\beta }V,T\rangle -\tfrac{1}{2}\langle j_{Z_\beta }U,V\rangle \langle j_{Z_\beta }S,T\rangle \bigr ) \nonumber \\&\quad \cdot \bigl (\tfrac{1}{4}\langle j_{Z_\gamma }T,X\rangle \langle j_{Z_\gamma }S,Y\rangle -\tfrac{1}{4}\langle j_{Z_\gamma }S,X\rangle \langle j_{Z_\gamma }T,Y\rangle -\tfrac{1}{2}\langle j_{Z_\gamma }S,T\rangle \langle j_{Z_\gamma }X,Y\rangle \bigr ) \end{aligned}$$

  (25)

  for $X,Y,U,V,S,T\in {\mathfrak v}$. For $a,b,c\in \{1,2,3\}$ denote by $(a*b*c)$ the sum over $\alpha ,\beta ,\gamma$ of the a-th summand in the first line, the b-th summand in the second, and the c-th summand of third line of (25). Then $(3*3*3)$ obviously yields, after summation over $X,Y,U,V,S,T\in \{X_1,\ldots ,X_m\}$, a multiple of $I_{\alpha \gamma |\beta \gamma |\alpha \beta }$, and thus contributes only to $L_2$. Five of the other $(a*b*c)$ (for example, $(1*1*1)$) lead to multiples of certain $I_{s_1\ldots s_6}$ in which the only subtuple is of length six (the reason being, just as we noted in the proof of (i), that there is no way to group the six factors into subsets which would not be linked to each other by the occurrence any common vectors from $\{X,Y,U,V,S,T\}$); these again contribute only to $L_2$. The only products which instead lead to a multiple of $I_{\alpha \beta \gamma |\alpha \beta \gamma }$ are $(1*1*2)$, $(1*2*1)$, $(2*1*1)$, and $(2*2*2)$. For example,

  $$\begin{aligned} (1*1*2)=-\tfrac{1}{64}\sum \nolimits _{\alpha ,\beta ,\gamma } \langle j_{Z_\alpha }Y,U\rangle \langle j_{Z_\beta }U,T\rangle \langle j_{Z_\gamma }T,Y\rangle \langle j_{Z_\alpha }X,V\rangle \langle j_{Z_\beta }V,S\rangle \langle j_{Z_\gamma }S,X\rangle \\ \end{aligned}$$

  which after summation gives

  $$\begin{aligned} -\tfrac{1}{64}\sum \nolimits _{\alpha ,\beta ,\gamma }{\text {Tr}}(j_{Z_\alpha }j_{Z_\beta }j_{Z_\gamma }){\text {Tr}}(j_{Z_\alpha }j_{Z_\beta } j_{Z_\gamma })=-\tfrac{1}{64}I_{\alpha \beta \gamma |\alpha \beta \gamma }. \end{aligned}$$

  The result is the same for each of the three other products just mentioned. So we obtain

  $$\begin{aligned} 4\cdot (-\tfrac{1}{64})I_{\alpha \beta \gamma |\alpha \beta \gamma }=-\tfrac{1}{16}I_{\alpha \beta \gamma |\alpha \beta \gamma } \end{aligned}$$

  as the contribution to $\hat{R}$ of the first summand in (24), apart from its contributions to $L_2$. For the second summand in (24), we compute

  $$\begin{aligned}&\langle R(X,Y)U,V\rangle \langle R(U,V)Z,W\rangle \langle R(Z,W)X,Y\rangle \nonumber \\&\quad = \sum \nolimits _{\alpha =1}^r \bigl (\tfrac{1}{4}\langle j_{Z_\alpha }Y,U\rangle \langle j_{Z_\alpha }X,V\rangle -\tfrac{1}{4}\langle j_{Z_\alpha }X,U\rangle \langle j_{Z_\alpha }Y,V\rangle -\tfrac{1}{2}\langle j_{Z_\alpha }X,Y\rangle \langle j_{Z_\alpha }U,V\rangle \bigr )\nonumber \\&\qquad \cdot \tfrac{1}{16}\langle [j_Z,j_W]U,V\rangle \langle [j_Z,j_W]X,Y\rangle . \end{aligned}$$

  (26)

  The first two summands from the first line, multiplied with the factors from the second line, will, after summation, yields multiples of certain $I_{s_1\ldots s_6}$ in which the only subtuple is of length six; this gives a contribution to $L_2$. The remaining term is

  $$\begin{aligned} -\tfrac{1}{32}\sum \nolimits _\alpha \langle j_{Z_\alpha }X,Y\rangle \langle j_{Z_\alpha }U,V\rangle \langle [j_Z,j_W]U,V\rangle \langle [j_Z,j_W]X,Y\rangle \end{aligned}$$

  which after summation over X, Y, U, V gives $-\tfrac{1}{32}\sum _\alpha ({\text {Tr}}(j_{Z_\alpha }[j_Z,j_W]))^2$; using skew-symmetry of the maps involved, this simplifies to $-\tfrac{1}{8}\sum _\alpha ({\text {Tr}}(j_{Z_\alpha }j_Zj_W))^2$. Summation over $Z,W\in \{Z_1,\ldots ,Z_r\}$ thus gives $-\tfrac{1}{8}I_{\alpha \beta \gamma |\alpha \beta \gamma }$. Hence, we obtain

  $$\begin{aligned} 3\cdot (-\tfrac{1}{8})I_{\alpha \beta \gamma |\alpha \beta \gamma } \end{aligned}$$

  as the contribution to $\hat{R}$ of the second summand in (24), apart from its contributions to $L_2$. The statement now follows by $-\frac{1}{16}-\frac{3}{8}=-\frac{7}{16}$.

• (iii) Although it would be possible to prove (iii) directly, similarly to the above proofs for (i) and (ii), we prefer to use the results of (i), (ii) together with those from Lemma 3 and the integral relation from Proposition 1(iii). If G(j) admits a compact quotient, then it follows from local homogeneity and Proposition 1(iii) that

  $$\begin{aligned} {\mathop R\limits ^{\circ }}=-|\nabla {\text {ric}}|^2+\tfrac{1}{4}|\nabla R|^2-{\text {Tr}}({\text {Ric}}^3)+(*)+\tfrac{1}{2}(**)-\tfrac{1}{4}\hat{R}. \end{aligned}$$

  From Lemma 1(iii) one easily derives ${\text {Tr}}({\text {Ric}}^3)=\frac{1}{8} I_{\alpha \alpha \beta \beta \gamma \gamma } -\frac{1}{64} I_{\alpha \beta |\beta \gamma |\gamma \alpha }$; thus, ${\text {Tr}}({\text {Ric}}^3)$ contributes to $L_3$ only. Now by (i), (ii) and Lemma 3, the right hand side is indeed of the form

  $$\begin{aligned} \tfrac{1}{4}\cdot (-\tfrac{3}{2})I_{\alpha \beta \gamma |\alpha \beta \gamma } -\tfrac{1}{4}\cdot (-\tfrac{7}{16})I_{\alpha \beta \gamma |\alpha \beta \gamma }+L_3 =-\tfrac{17}{64}I_{\alpha \beta \gamma |\alpha \beta \gamma }+L_3, \end{aligned}$$

  where $L_3$ is a linear combination of invariants in which only subtuples of even length occur. So we have proved the statement of (iii) in the case that G(j) admits a compact quotient. The statement in the general case now follows by continuity. In fact, any G(j) for which j is a rational map w.r.t. the standard rational structures on ${\mathfrak z}={\mathbb R}^r$ and ${\mathfrak {so}}({\mathfrak v})={\mathfrak {so}}(m)$ does admit a compact quotient, and the rational maps are dense in the space of all linear maps $j:{\mathfrak z}\rightarrow {\mathfrak {so}}({\mathfrak v})$. $\square$