Submanifolds under affine and Riemannian Structures

The present project proposes a study of submanifolds in the setting of Riemannian and affine differential geometry. In submanifold theory,
one studies manifolds which are immersed -as submanifolds- in other manifolds -called ambient spaces.
The geometric structures carried by the ambient spaces define their geometry and influence the existence of certain classes of
submanifolds. The general purpose of the project is to obtain an understanding of the nearly Kaehler manifold S^3 x S^3
and R^n endowed with an affine differential structure, through the geometric data encoded in a few classes of their submanifolds.
 To do so, we aim at constructing new examples of submanifolds, give existence results, characterizations and classification theorems.
The project comprises four problems. The first three are related to the nearly Kaehler S^3 x S^3, on which the developments are very
recent. Following some previous results of mine, I want to tackle three new problems concerning conformally flat Lagrangian submanifolds
CR submanifolds and compact Lagrangian submanifolds, respectively.
Finally, in the fourth problem I will study hypersurfaces of the affine space R^n that are of warped product type.