Given a fixed metric $g_{\mu\nu}$, its variation by a small amount could be written as: $$g_{\mu\nu}+h_{\mu\nu}$$ or equivalently as: $$g_{\mu\nu}+\delta(g_{\mu\nu}).$$ The given metric has the property that: $$g_{\mu\alpha}g^{\alpha\nu}=\delta^\nu_\mu,$$ and that $g_{\mu\nu}=g_{\nu\mu}$, which applies also to the metric after variation $g+h$.

Question: What is the relationship between $\delta g^{\mu\nu}$ and $h^{\mu\nu}$? This is actually proving that for arbitrary tensors: $$A_{\mu\nu}B^{\mu\nu}=A^{\mu\nu}B_{\mu\nu}.$$ How to prove this?