${\mathbb C}$, respectively. Suppose that the functions $F^1,\ldots,F^N$ don’t depend on the variable $t^{N+1}$, varying in $U$. Then the functions $F^1,\ldots,F^N$ satisfy equations  and, thus, define a flat F-manifold structure on $M$. In this case we call the flat F-manifold structure on $M\times U$ an [*extension*]{} of the flat F-manifold structure on $M$.

Consider the flat F-manifold associated to a Frobenius manifold, given by a potential $F(t^1,\ldots,t^N)\in{\mathcal{O}}(M)$ and a metric $\eta$, $F^\alpha=\eta^{\alpha\mu}\frac{\d F}{\d t^\mu}$, $1\le\alpha\le N$. It is easy to check that a function $F^o(t^1,\ldots,t^N,s)\in{\mathcal{O}}(M\times U)$ satisfies equations ,  and  if and only if the $(N+1)$-tuple $(F^1,\ldots,F^N,F^o)$ is a vector potential of a flat F-manifold. Here we identify $s=t^{N+1}$. This defines a correspondence between solutions of the open WDVV equations, satisfying property , and flat F-manifolds, extending the Frobenius manifold given. This observation belongs to Paolo Rossi.

Saito Frobenius manifolds
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In this section we recall the Dubrovin–Saito construction of a Frobenius manifold structure on the parameter space of a universal unfolding of a simple singularity.

Let us first recall the list of polynomials defining simple singularities: $$\begin{aligned}
f_{A_N}(x,y)=&\frac{x^{N+1}}{N+1} + y^2,&& N\ge 1,\\
f_{D_N}(x,y)=&\frac{x^{N-1}}{N-1} + x y^2,&& N\ge 4,\\
f_{E_6}(x,y)=&x^4 + y^3,\\
f_{E_7}(x,y)=&x^3y + y^3,\\
f_{E_8}(x,y)=&x^5 + y^3.\end{aligned}$$ The associated 