Frobenius manifolds and flat pencils of metrics

This thesis investigates Frobenius manifolds from a differential geometric perspective. We begin by employing tools from Riemannian geometry to establish the equivalence between Frobenius manifolds and flat pencils of metrics. In this context, we analyze the Levi-Civita connection, the Euler and unity vector fields, and the potential function appearing in the structure equations. Concrete examples are provided to illustrate the emergence of this structure, including explicit computations of the associated metrics and bihamiltonian systems. Finally, we extend the Frobenius manifold framework to supermanifolds and compute the explicit form of the bihamiltonian system in the (3,2) case.