Kulkarni-Nomizu identity

Question

I am trying to prove the identity $\newcommand\KN{\bigcirc \kern-2.5ex\wedge \;}$

$$ g(T,h \KN\, g)=4g({\rm tr}_g T,h) $$

where $h$ is a symmetric tensor of type $(0,2)$, $T$ is an algebraic curvature tensor and $g$ is a metric for a vector space $V$. Here, $h \KN\, g$ denotes the Kulkarni-Nomizu product between $h$ and $g$. In my attempt, I wrote in coordinates the left part of the equality:

$$ T_{ijkl}(h_{mp}g_{no}+h_{no}g_{mp}-h_{mo}g_{np}-h_{np}g_{mo})g^{im}g^{jk}g^{ko}g^{lp} $$ using the formula for the inner product of covariant tensors, but I don't get anywhere. Any help?

Answer 1

$\newcommand\KN{\bigcirc \kern-2.5ex\wedge \;}$The idea is to repeatedly use the relation ${\rm tr}_g(T)_{jk} = T_{ijk\ell}g^{i\ell}$, paying attention to the role of dummy indices to recognize more occurence. This way, we have that

$$\begin{split} g(T, h \KN \,g) &=T_{ijk\ell} (h \KN \,g)^{ijk\ell} \\ &= T_{ijk\ell}(h^{jk}g^{i\ell} - h^{ik}g^{j\ell} + g^{jk}h^{i\ell} - g^{ik}h^{j\ell}) \\ &= T_{ijk\ell}h^{jk}g^{i\ell} - T_{ijk\ell} h^{ik}g^{j\ell} + T_{ijk\ell}g^{jk}h^{i\ell} - T_{ijk\ell}g^{ik}h^{j\ell} \\ &\stackrel{(\ast)}{=} (T_{ijk\ell}g^{i\ell})h^{jk} + (T_{jik\ell}g^{j\ell})h^{ik}+(T_{ji\ell k}g^{jk})h^{i\ell} + (T_{ij\ell k}g^{ik})h^{j\ell} \\ &= {\rm tr}_g(T)_{jk}h^{jk} + {\rm tr}_g(T)_{ik}h^{ik} + {\rm tr}_g(T)_{i\ell}h^{i\ell} + {\rm tr}_g(T)_{j\ell}h^{j\ell} \\ &= g({\rm tr}_g(T), h)+g({\rm tr}_g(T), h)+g({\rm tr}_g(T), h)+g({\rm tr}_g(T), h) \\ &= 4g({\rm tr}_g(T), h),\end{split}$$as required. The crucial step in this computation is $(\ast)$, where several curvature symmetries of $T$ are used. Namely:

• Nothing happens on the first term $T_{ijk\ell}$.
• Skew-symmetry on the first pair $-T_{ijk\ell} = T_{jik\ell}$ is used on the second term.
• Skew-symmetry on both pairs $T_{ijk\ell} = T_{ji\ell k}$ is used on the third term.
• Skew-symmetry on the second pair $-T_{ijk\ell} = T_{ij\ell k}$ is used on the fourth term.