Some negative property of Riemann curvature

Question

I am trying to consider a question about the following, suppose $T$ is any symmetric trace-free 2-tensor, what conditions on curvature are sufficient for $$T^{ij}T^{kl}R_{ikjl}\leq 0\,?$$ My definition of curvature is $$R(X,Y)=\nabla_Y\nabla_X-\nabla_X\nabla_Y+\nabla_{[X,Y]}$$ $$R_{ikjl}=R(e_i,e_k,e_j,e_l)=\langle R(e_i,e_j)e_k,e_l\rangle $$ Note that when $R_{ikjl}=g_{ij}g_{kl}-g_{il}g_{kj}$, we have $T^{ij}T^{kl}R_{ikjl}=-T^{ij}T_{ij}\leq 0$. Does there exists a more general type of manifold has this negative property?