Show that $\partial_t g^{jl}=-g^{jp}(\partial_tg_{pq})g^{ql}$

Question

If $g_{ij}$ are the components of a Riemannian metric and $\partial_tg_{ij}=-2R_{ij}$, how to show

$$\partial_t g^{jl}=-g^{jp}(\partial_tg_{pq})g^{ql}\,?$$

Answer 1

Note that

$$\delta_p^l = g_{pq}g^{ql}.$$

Differentiating with respect to $t$ we get

$$0 = (\partial_tg_{pq})g^{ql} + g_{pq}(\partial_tg^{ql})$$

so

$$g_{pq}(\partial_t g^{ql}) = - (\partial_t g_{pq})g^{ql}.$$

Applying $g^{jp}$ to both sides gives

$$g^{jp}g_{pq}(\partial_t g^{ql}) = - g^{jp}(\partial_t g_{pq})g^{ql}.$$

As $g^{jp}g_{pq} = \delta^j_q$, we have

$$\partial_tg^{jl} = - g^{jp}(\partial_t g_{pq})g^{ql}.$$