Realization of Frobenius Manifolds as Submanifolds in Pseudo-Euclidean Spaces

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.