Marie Skłodowska-Curie grant agreement No. 797635. The first named author was supported by RSF Grant No. 19-71-00086.

Flat F-manifolds and Frobenius manifolds {#section:definitions}
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In this section we recall the definitions and the main properties of flat F-manifolds and Frobenius manifolds. We also explain how solutions of the open WDVV equations correspond to flat F-manifolds of special type.

Flat F-manifolds {#subsection:flat F-manifolds}
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The notion of a flat F-manifold was introduced in [@Man05].

A flat F-manifold $(M,\nabla,\circ)$ is the datum of a complex analytic manifold $M$, an analytic connection $\nabla$ in the tangent bundle $T M$, an algebra structure $(T_p M,\circ)$ with unit $e$ on each tangent space analytically depending on the point $p\in M$ such that the one-parameter family of connections $\nabla+z\circ$ is flat and torsionless for any $z\in{\mathbb C}$, and $\nabla e=0$.

For a flat F-manifold $(M,\nabla,\circ)$ consider flat coordinates $t^\alpha$, $1\le\alpha\le N$, $N=\dim M$, for the connection $\nabla$ such that $e = \frac{\d}{\d t^1}$. Then locally there exist analytic functions $F^\alpha(t^1,\ldots,t^N)$, $1\leq\alpha\leq N$, such that the second derivatives $$\begin{gathered}
\label{eq:structure constants of flat F-man}
c^\alpha_{\beta\gamma}=\frac{\d^2 F^\alpha}{\d t^\beta \d t^\gamma}\end{gathered}$$ give the structure constants for the multiplication $\circ$, $$\begin{gathered}
\frac{\d}{\d t^\beta}\circ\frac{\d}{\d t^\gamma}=c^\alpha_{\beta\gamma}\frac{\d}{\d t^\alpha}.\end{gathered}$$ From the associativity of the 