Flat pencils of metrics and Frobenius manifolds

TL;DR

This paper explores the geometric relationship between flat pencils of metrics and Frobenius manifolds, revealing their equivalence under certain conditions and their role in classifying bihamiltonian structures in integrable systems.

Contribution

It establishes that flat pencils of contravariant metrics and Frobenius manifolds are equivalent under homogeneity assumptions, linking geometric structures to integrable hierarchies.

Findings

Flat pencils of metrics are naturally related to Frobenius manifolds.

Under certain conditions, these two structures are shown to be identical.

The work clarifies the connection between Frobenius manifolds and integrable hierarchies.

Abstract

This paper is based on the author's talk at 1997 Taniguchi Symposium ``Integrable Systems and Algebraic Geometry''. We consider an approach to the theory of Frobenius manifolds based on the geometry of flat pencils of contravariant metrics. It is shown that, under certain homogeneity assumptions, these two objects are identical. The flat pencils of contravariant metrics on a manifold $M$ appear naturally in the classification of bihamiltonian structures of hydrodynamics type on the loop space $L(M)$. This elucidates the relations between Frobenius manifolds and integrable hierarchies.