Let $\Sigma$ be a hypesurface immersed in a Riemannian manifold $M.$ Suppose that $h$ is the second fundamental form of $\Sigma.$
Now, let $\sigma_p$ denote the $p$-th elementary symmetric polynomial in the eigenvalue of $h.$ What does the notation $\frac{\partial\sigma_p}{\partial h_{ij}}$ mean?
In the paper https://arxiv.org/abs/1208.3988v2, this notation provides a symmetric $2$-tensor. However, it is not clear to me. For example, what is $\frac{\partial\sigma_1}{\partial h_{ij}}$?