In recent years due to the work of amongst others Butruille, Spiro, Podesta and Nagy a considerable amount of progress has been made in the study and classification of nearly Kaehler maniolfds. According to Nagy’s structure theorem any complete strict nearly kaeher manifol is finitely covered by a product of homogeneous 3-symmetric manifolds, twistor spaces of positive quaternion Kaehler manifolds with their canonical NK structur and six dimensional strict NK manifolds. This is one of the reasons which raise a particular interest for six dimensional strict NK structures. It is also known that, in six dimensions, the “strictness” condition is equivalent to the fact that the NK structure is not Kaehler and that strict NK manifolds are automatically Einstein and related with the existence of a nonzero Killing spinor.Other reasons of interest for NK structures in six dimensions are provided by their relations with geometries with torsion, G2-holonomy and supersymmetric models. The only homogeneous strict NK manifolds in six dimensions are the six dimensional 3-symmetric spaces endowed with their natural NK structures, namely the standard sphere $S^6 = G2/SU3$, the twistor spaces $ CP2 = Sp2/U(1) \times Sp1$ and $F = SU3/U(1)^2$ and the space $S^3 \times S^3$. Whereas submanifolds of $S^6$ are well understood by now, this is not yet the case for submanifolds of $S^3\times S^3$ (with respect to this nearly Kaehler structure). Not that the metric associated with this structure is not the standard metric on $S^3 \times S^3$. The aim of this lecture is to present the structure in an elementary way which will allow the systematic study of its submanifolds. We will then focus on almost complex curves for which we will introduce a holomorphic differential. Further results include a classification of all totally geodesic almost complex curves as well as as the result that an almost complex $S^2$ is totally geodesic.

A surface is called a constant angle surface if its unit normal makes a constant angle with a "fixed" direction. Starting from this general definition, we give different choices of directions in order to study and classify constant angle surfaces. Firstly, in the Euclidean 3-space, one of the coordinates axes, the position vector field, or a Killing vector field may be chosen as a "fixed" direction. Secondly, recent classification results for constant angle surfaces in solvable Lie groups are obtained.