Realization of Frobenius manifolds as submanifolds in pseudo-Euclidean spaces

TL;DR

This paper demonstrates that any N-dimensional Frobenius manifold can be locally embedded as a k-potential submanifold in a suitably dimensioned pseudo-Euclidean space, generalizing previous work for k=1.

Contribution

It introduces k-potential submanifolds and proves their ability to realize all Frobenius manifolds in higher-dimensional pseudo-Euclidean spaces.

Findings

Any Frobenius manifold can be realized as a k-potential submanifold.

Realization reduces to solving a linear PDE system.

Construction extends previous k=1 case to arbitrary k and p.

Abstract

We introduce a class of k-potential submanifolds in pseudo-Euclidean spaces and prove that for an arbitrary positive integer k and an arbitrary nonnegative integer p, each N-dimensional Frobenius manifold can always be locally realized as an N-dimensional k-potential submanifold in ((k + 1) N + p)-dimensional pseudo-Euclidean spaces of certain signatures. For k = 1 this construction was proposed by the present author in a previous paper (2006). The realization of concrete Frobenius manifolds is reduced to solving a consistent linear system of second-order partial differential equations.