Frobenius Manifolds: Natural submanifolds and induced bi-Hamiltonian structures

Abstract

Submanifolds of Frobenius manifolds are studied. In particular, so-called natural submanifolds are defined and, for semi-simple Frobenius manifolds, classified. These carry the structure of a Frobenius algebra on each tangent space, but will, in general, be curved. The induced curvature is studied, a main result being that these natural submanifolds carry a induced pencil of compatible metrics. It is then shown how one may constrain the bi-Hamiltonian hierarchies associated to a Frobenius manifold to live on these natural submanifolds whilst retaining their, now non-local, bi-Hamiltonian structure.

27 Pages, LaTeX, 1 figure

References in corpus (1)

• Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov - Witten invariants

Cited by in corpus (22)

• Local Moduli of Semisimple Frobenius Coalescent Structures
• Dubrovin's duality for $F$-manifolds with eventual identities
• $F$-manifolds with eventual identities, bidifferential calculus and twisted Lenard-Magri chains
• Weyl groups and Elliptic Solutions of the WDVV equations
• Compatible metrics on a manifold and non-local bi-Hamiltonian structures
• A note on the relationship between rational and trigonometric solutions of the WDVV equations
• Infinite hierarchies of nonlocal symmetries of the Chen--Kontsevich--Schwarz type for the oriented associativity equations
• Generalized Legendre transformations and symmetries of the WDVV equations
• Frobenius manifolds and Frobenius algebra-valued integrable systems
• Classical $r$-matrix like approach to Frobenius manifolds, WDVV equations and flat metrics
• Duality for Jacobi group orbit spaces and elliptic solutions of the WDVV equations
• Coxeter discriminants and logarithmic Frobenius structures
• Isomonodromic deformations along a stratum of the coalescence locus
• The Differential Geometry of the Orbit Space of Extended Affine Jacobi Group $A_1$
• Logarithmic deformations of the rational superpotential/Landau-Ginzburg construction of solutions of the WDVV equations
• Extended V-systems and almost-duality for extended affine Weyl orbit spaces
• Somonodromic Deformations Along the Caustic of a Dubrovin-Frobenius Manifold
• Modular Frobenius manifolds and their invariant flows
• (TE)-structures over the irreducible 2-dimensional globally nilpotent F-manifold germ
• Formal Frobenius structures generated by geometric deformation algebras
• The Saito determinant for Coxeter discriminant strata
• Diagonal invariants and genus-zero Hurwitz Frobenius manifolds