Non-flat Frobenius manifolds: geometry and integrability

We introduce the notion of a non-flat Frobenius manifold, which generalizes the notion of a flat Dubrovin–Frobenius manifold. Locally, non-flat Frobenius manifolds are described by the curved Witten–Dijkgraaf–Verlinde–Verlinde equations (the curved WDVV equations) naturally arising in some physical models and in the theory of submanifolds with potental of normals in pseudo-Euclidean spaces that was developed by the author. Earlier it was proved by the author that the WDVV equations are natural special reductions of the fundamental equations of the theory of submanifolds in pseudo-Euclidean spaces and any Dubrovin–Frobenius manifold can be realized as a special flat submanifold with flat normal bundle in a pseudo-Euclidean space. In this talk we prove that the curved WDVV equations are also natural special reductions of the fundamental equations of the theory of submanifolds in pseudo-Euclidean spaces and any non-flat Frobenius manifold can be realized as a special submanifold with potential of normals in a pseudo-Euclidean space. Besides, we propose a Lax pair for the curved WDVV equations and claim that the curved WDVV equations are integrable.
This research was supported by the Russian Science Foundation under grant 20-11-20214.