If $g_t$ is a time dependent metric on $M$ and $\nabla$ is connection, if $Y$ is a smooth vector field on $M$, then why does $\nabla_{Y} \partial_t g_t$ makes sense?
I don't understand why $\partial_t g_t$ is a tensor.
Namely, I know that if $T$ is a $(a,b)$ Tensor defined on $M$, and $Y$ is a smooth vector field on $M$, then $\nabla_{Y}T$ makes sense and has an explicit formula.