On adding a variable to a Frobenius manifold and generalizations

Let \pi : V \rightarrow M be a (real or holomorphic) vector bundle whose base has an almost Frobenius structure (\circ_{M},e_{M}, g_{M}) and typical fiber has the structure of a Frobenius algebra (\circ_{V},e_{V},g_{V}). Using a connection D on the bundle V and a morphism \alpha : V \rightarrow TM, we construct an almost Frobenius structure (\circ,e_{V},g) on the manifold V and we study when it is Frobenius. We describe all (real) positive-definite Frobenius structures on V, obtained in this way, when M is a semisimple Frobenius manifold with non-vanishing rotation coefficients. In the holomorphic setting we add a real structure k_{M} on M and a real structure k_{V} on the fibers of \pi and we study when an induced real structure on the manifold V, together with the almost Frobenius structure (\circ, e_{V}, g), satisfy the tt*-equations. Along the way, we prove various properties of adding variables to a Frobenius manifold, in connection with Legendre transformations and tt*-geometry.