Curved Witten-Dijkgraaf-Verlinde-Verlinde equation and $\mathcal{N}=4$ mechanics

We propose a generalization of the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equation from ${\mathbb{R}}^n$ to an arbitrary Riemannian manifold. Its form is obtained by extending the relation of the WDVV equation with $\mathcal{N}=4$ supersymmetric $n$-dimensional mechanics from flat to curved space. The resulting “curved WDVV equation” is written in terms of a third-rank Codazzi tensor. For every flat-space WDVV solution subject to a simple constraint, we provide a curved-space solution on any isotropic space, in terms of the rotationally invariant conformal factor of the metric.