I have an $n-$dimensional manifold. On the manifold, I have a set of 1-forms L which depend parametrically on a "time" variable. In other words, in a coordinate patch $L(t)=u_i(t,x^1,\ldots,x^n)dx^i$ with $t>=0$. Starting from a given metric tensor at $t=0$, I evolve it in time according to $$ \frac{\partial g}{\partial t}=\widetilde{\nabla_t L(t)} $$ where the tilde represents the symmetric part of its argument, and $\nabla_t$ is the covariant derivative with the Levi-Civita connection of the metric $g(t)$. In a coordinate patch $$ \frac{\partial g_{ij}}{\partial t}=\left(\frac{\partial u_i}{\partial x^j}+\frac{\partial u_j}{\partial x^i}-2u_p\Gamma^p_{ij}(t)\right) $$
I want the Riemann curvature tensor to remain constant in time, set by the curvature of the initial metric $g(0)$. Do I need to impose additional conditions on $L(t)$?