On adding a variable to a Frobenius manifold and generalizations

Let $pi :Vrightarrow M$ 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 $pi : V{,rightarrow ,}M$ and a morphism $alpha :Vrightarrow TM$ , we construct an almost Frobenius structure $(circ , e_{V},g)$ on the manifold $V$ and we study when it is Frobenius. In particular, 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 bundle $pi : V rightarrow M$ . Using $k_{M}$ , $k_{V}$ and $D$ we define a real structure $k$ on the manifold $V$ . We study when $k$ , together with an almost Frobenius structure $(circ , e_{V}, g) $ , satisfies 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.