Meromorphic connections over F-manifolds

Abstract

This paper review one construction of Frobenius manifolds (and slightly weaker structures). It splits it into several steps and discusses the freedom and the constraints in these steps. The steps pass through holomorphic bundles with meromorphic connections. A conjecture on existence and uniqueness of certain such bundles, a proof of the conjecture in the 2-dimensional cases, and some other new results form a research part of this paper. Contents 42 8. (T E)-structures over the 2-dimensional F-manifolds I 2 (m) 46 References 55

Riemannian F-manifolds, bi-flat F-manifolds, and flat pencils of metrics

“…(1) Dropping the axioms involving explicitly the metric apart from those involving only the Levi-Civita connection in the definition of a (conformal) Dubrovin-Frobenius manifold. This is the most straightforward way and leads to the definition of a Dubrovin-Frobenius manifold without metric [AL17] or flat F-manifold with (linear) Euler vector field [DH19] or homogeneous flat F-manifold [ABLR20a] (see also [ABLR20b, Remark 2.2]).…”