On $R^n$ consider distance function $r(x)=|x|$ and metric $g=dr^2+g_{r}=dr^2+\varphi(r)^2ds_{n-1}^2$, where $ds_{n-1}^2$ is the canonical metric on $S^{n-1}(1)$. Then the directional differential $\triangledown_{X}g_r=0$, no matter what the vector field $X$ is.
Since $g_r$ is a tensor we have
$\triangledown_{X}g_r(U,V)=\triangledown_{X}(g_r(U,V))-g_r(\triangledown_{X}U,V)-g_r(U,\triangledown_{X}V)$
for any vector field on $S^{n-1}(r)$. But it seems the equation can't be zero all the time. What do I miss?