an analytic function $F$ such that $F^\alpha = \eta^{\alpha\beta}\frac{\d F}{\d t^\beta}$ and $\frac{\d^3 F}{\d t^1\d t^\alpha\d t^\beta}=\eta_{\alpha\beta}$, where $(\eta_{\alpha\beta})$ is the matrix of the form $\eta$ in the coordinates $t^1,\ldots,t^N$. The function $F$ satisfies the WDVV equations  and is called the Frobenius manifold potential.

A Frobenius manifold $(M,\eta,\circ)$ is called conformal if the corresponding flat F-manifold is conformal and the metric $\eta$ satisfies the condition $${\mathcal{L}}_E\eta=(2-\delta)\eta,\quad\text{for some $\delta\in{\mathbb C}$},$$ where ${\mathcal{L}}_E$ denotes the Lie derivative. The number $\delta$ is then called the conformal dimension of the Frobenius manifold. The Frobenius manifold potential $F$ satisfies then the condition $$E(F) = (3-\delta) F + \frac{1}{2}A_{\alpha\beta}t^\alpha t^\beta+B_\alpha t^\alpha+C,\quad\text{for some $A_{\alpha\beta},B_\alpha,C\in{\mathbb C}$}.$$ In the theory of Frobenius manifolds it is typically assumed that one can choose flat coordinates such that the matrix $\left(\frac{\d E^\alpha}{\d t^\beta}\right)$ is diagonal and so the Euler vector field has form .

The papers [@Dub96; @Dub99] contain a systematic study of the theory of Frobenius manifolds.

Extensions of flat F-manifolds and the open WDVV equations {#subsection:flat F-manifolds and open WDVV}
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Consider a flat F-manifold structure, given by a vector potential $(F^1,\ldots,F^{N+1})$ on an open subset ${M\times U\in{\mathbb C}^{N+1}}$, where $M$ and $U$ are open subsets of ${\mathbb C}^N$ and 