multiplication and the fact that the vector field $\frac{\d}{\d t^1}$ is the unit it follows that $$\begin{aligned}
\frac{\d^2 F^\alpha}{\d t^1\d t^\beta} &= \delta^\alpha_\beta, && 1\leq \alpha,\beta\leq N,\label{eq:axiom1 of flat F-man}\\
\frac{\d^2 F^\alpha}{\d t^\beta \d t^\mu} \frac{\d^2 F^\mu}{\d t^\gamma \d t^\delta} &= \frac{\d^2 F^\alpha}{\d t^\gamma \d t^\mu} \frac{\d^2 F^\mu}{\d t^\beta \d t^\delta}, && 1\leq \alpha,\beta,\gamma,\delta\leq N.\label{eq:axiom2 of flat F-man}\end{aligned}$$ The $N$-tuple of functions $(F^1,\ldots,F^N)$ is called the [*vector potential*]{} of our flat F-manifold.

Conversely, if $M$ is an open subset of ${\mathbb C}^N$ and $F^1,\ldots,F^N\in{\mathcal{O}}(M)$ are functions satisfying equations  and , then these functions define a flat F-manifold $(M,\nabla,\circ)$ with the connection $\nabla$, given by $\nabla_{\frac{\d}{\d t^\alpha}}\frac{\d}{\d t^\beta}=0$, and the multiplication $\circ$, given by the structure constants .

\[remark:algebraic construction of multiplication\] Let $M\subset{\mathbb C}^N$ be an open subset in the Zariski topology. The tangent spaces $T_p M$ can be endowed with an algebra structure, algebraically depending on the point $p\in M$, using the following construction. Denote by ${\mathcal{O}}^{\mathrm{alg}}$ the sheaf of algebraic functions on $M$. Let $R$ be an ${\mathcal{O}}^{\mathrm{alg}}(M)$-algebra, which is free as an ${\mathcal{O}}^{\mathrm{alg}}(M)$-module with a basis $\phi_1,\ldots,\phi_N\in R$. Denote by $v_1,\ldots,v_N$ the standard coordinates on ${\mathbb C}^N$ and by ${\mathcal{T}}_M^{\mathrm{alg}}$ the algebraic tangent sheaf of $M$. Define 