Question: Does there exist a Helmholtz-like decomposition of the metric tensor into holonomic and anholonomic counterparts?
Motivation: I know that for a given tensor function $F$, one can find a vector-valued function $u$ and a tensor-valued function $A$ such that $F=Du+\text{Curl}A$. In this case, the requirement $\text{Curl}F=0$ is an integrability condition for $F$.
What I wonder is if an analogous conclusion holds for a metric tensor $G$. In this case, the integrability condition is $\text{Riem}(G)=0$. When this is satisfied, there exists a vector-valued function $u$ such that $G=(Du)^T(Du)$.
But what if $\text{Riem}(G)\neq 0$? For given $G$, would there exist a vector valued function $u$ and a (symmetric) tensor function $A$ such that $$ G=(Du)^T(Du) + \text{Riem}(A)\qquad (\text{or similar}?) $$