Research On Natural Frobenius Submanifolds And Tt~* Geometry

Frobenius manifolds were first introduced by Dubrovin to explain 2D topologicaltheory. Strachan introduced the notation of natural Frobenius sub-manifolds, and givea sufficient but not necessary condition for an embedding sub-manifold to be a naturalFrobenius sub-manifold.We will study the geometry of a sub-manifold of a given Frobenius manifold, andobtain a sufficient and necessary condition for a sub-manifold to be a natural Frobeniussub-manifold. Our result covers Strachan's condition. We also apply this result to thecase of hyper-surfaces and obtain a classification of natural Frobenius hyper-surfaces.The second topic in this paper is the geometry of a CDV -structure. This structurewas first introduced by C. Hertlling. C. Hertling also gave a equivalent definition start-ing from a CV -structure. In this paper we will give a sufficient and necessary conditionfor a Frobenius manifold to be a CDV -structure. This condition simplifies the originaldefinition very much. We have known a lot of examples of Frobenius manifolds, so thiscondition will play an important role in constructing examples in CDV -structure in thefuture.We finally study the special case: if a Frobenius manifold of a CDV -strucure issemi-simple, and the two structure and D in CDV -structure are equal to each other,then the canonical local coordinates of the Frobenius manifold is ?at, in particular, theFrobenius manifold is trivial.