an isomorphism of ${\mathcal{O}}^{\mathrm{alg}}(M)$-modules $\Psi\colon{\mathcal{T}}_M^{\mathrm{alg}}(M)\to R$ by $\Psi(\frac{\d}{\d v_i}):=\phi_i$. Thus, the sheaf ${\mathcal{T}}_M^{\mathrm{alg}}$ becomes a sheaf of ${\mathcal{O}}^{\mathrm{alg}}$-algebras, that endows the tangent spaces $T_p M$ with an algebra structure algebraically depending on the point $p\in M$.

Consider an analytic manifold $M$ with an algebra structure $(T_p M,\circ)$ on each tangent space analytically depending on the point $p\in M$. We see that a connection $\nabla$, endowing our manifold $M$ with a flat F-manifold structure, can be completely described by a choice of coordinates $t^1,\ldots,t^N$ on $M$ such that the structure constants $c^\alpha_{\beta\gamma}$ of multiplication in these coordinates satisfy the integrability condition $$\frac{\d c^\alpha_{\beta\gamma}}{\d t^\sigma}=\frac{\d c^\alpha_{\beta\sigma}}{\d t^\gamma}$$ together with the condition $c^\alpha_{1,\beta}=\delta^\alpha_\beta$. In this paper we will construct flat $F$-manifolds exactly by presenting flat coordinates as above.

A flat F-manifold $(M,\nabla,\circ)$ is called [*conformal*]{}, if it is equipped with a vector field $E$, called the [*Euler vector field*]{}, such that $\nabla\nabla E=0$, $[e,E]=e$ and ${\mathcal{L}}_E(\circ)=\circ$. This means that in the flat coordinates the Euler vector field $E$ has the form $$E=\underbrace{(q^\alpha_\beta t^\beta+r^\alpha)}_{=:E^\alpha}\frac{\d}{\d t^\alpha},\quad q^\alpha_\beta,r^\alpha\in{\mathbb C},\quad q^\alpha_1=\delta^\alpha_1,$$ and the vector potential $(F^1,\ldots,F^N)$ satisfies the condition $$\begin{gathered}
\label{eq:homogeneity for F-man}
E^\mu\frac{\d F^\alpha}{\d t^\mu}=q^\alpha_\beta F^\beta+F^\alpha+A^\alpha_\beta t^\beta+B^\alpha,\quad\text{for some $A^\alpha_\beta,B^\alpha\in{\mathbb C}$}.\end{gathered}$$

A point $p\in M$ of an 