Compatible metrics on a manifold and nonlocal Bi-hamiltonian structures

David, L. and Strachan, I.A.B. (2004) Compatible metrics on a manifold and nonlocal Bi-hamiltonian structures. International Mathematics Research Notices, 66, pp. 3533-3557. (doi: 10.1155/S1073792804142359)

Abstract

Given a flat metric, one may generate a local Hamiltonian structure via the fundamental result of Dubrovin and Novikov. More generally, a flat pencil of metrics will generate a local bi-Hamiltonian structure, and with additional quasihomogeneity conditions, one obtains the structure of a Frobenius manifold. With appropriate curvature conditions, one may define a curved pencil of compatible metrics and these give rise to an associated nonlocal bi-Hamiltonian structure. Specific examples include the F-manifolds of Hertling and Manin equipped with an invariant metric. In this paper, the geometry supporting such compatible metrics is studied and interpreted in terms of a multiplication on the cotangent bundle. With additional quasihomogeneity assumptions, one arrives at a so-called weak Formula-manifold, a curved version of a Frobenius manifold (which is not, in general, an F-manifold). A submanifold theory is also developed.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Strachan, Professor Ian
• Authors: David, L., and Strachan, I.A.B.
• Subjects: Q Science > QA Mathematics
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: International Mathematics Research Notices
• ISSN: 1073-7928
• ISSN (Online): 1687-0247