Let $M$ be a smooth manifold, let $I \subset \mathbb{R}$ be an open interval, and let $(g(t))_{t \in I}$ be a smooth one-parameter family of Riemannian metrics on $M$. For each $t \in I$, let $\nabla(t)$ be the Levi-Civita connection of $g(t)$, and define the symmetric $(0,2)$-tensor