Hi all,

I have a question of the following form: Let $(M,g)$ be a Riemannian spin manifold which admits a Killing spinor $\sigma$ and let $h:T M \to T M$ be a symmetric, trace-free and divergence-free tensor. Consider the following expression: $\sum_{i=1}^n\nabla_i h(Y)\cdot\nabla_i\sigma$

The question is: can this expression be zero for every $Y$ without $\nabla h$ being zero?

In fact, this is equivalent to the following equation: $\Delta(h(Y)\cdot\sigma)=(\Delta h(Y)+c^2 n h(Y))\cdot\sigma$

$\Delta$ is the usual Laplace Beltrami operator.

I am thankful for any help.

"Edit": I translated the question in a probably more considerable form: I am looking for trace-free and divergence-free symmetric tensors which are in the kernel of the generalized exterior derivative: $d h(X,Y)=\nabla_X h(Y)-\nabla_Y h(X)$