Trace of hessian

Question

So this is a result used in Peter Topping's,Lectures on Ricci Flow. What is a quick of showing

$$\text{tr}\nabla_{X,\cdot}^2h(\cdot,W)=-(\nabla\delta h)(X,W)$$

where $\delta A=-\text{tr}_{12}\nabla A$, is the divergence operator and $h=\partial_tg$.

Answer 1

This is just raising/lowering the indices in the trace - the particular value of $h$ is irrelevant. In abstract index notation we have

$${\rm tr} \nabla_{X,\cdot}^2 h(\cdot,W) = \nabla_X \nabla_j h^j{}_W = \nabla_X \nabla^j h_{jW} = \nabla_X (-\delta h)(W)=-(\nabla\delta h)(X,W).$$

More abstractly, just note that the RHS is $-\nabla (-{\rm tr}_{12} \nabla h)$ which is ${\rm tr}_{23} \nabla^2 h$ because contractions commute with the covariant derivative, and we have to change the slots we are contracting over because the $\nabla$ adds an extra slot at the front.