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 tensor T is called isotropic if T(v,...,v) is independent of the choice of unit length vector v. The first result about Lagrangian submanifolds admitting an isotropic tensor was due to Naitoh who classified isotropic parallel Lagrangian submanifolds of complex space forms. Later results are due to Montiel, Urbano and Ejiri. In both cases the tensor T(X,Y,Z,W)=< h(X,Y),h(Z,W) >, where h is the second fundamental form of the Lagrangian immersion. The classification result in this case either gives the parallel hypersurfaces of Naitoh or a special class of H umbilical Lagrangian submanifolds. Of course the same question can also be asked for other geometric tensors like T(X,Y,Z,W)=<∇h(X,Y,Z),JW> or T(X,Y,Z,W,U,V)= <∇h(X,Y,Z),∇h (W,U,V)>. The first condition actually can be used to characterize the Whitney spheres, together with the parallel Lagrangian submanifolds. Whereas the second one is more difficult to treat and so far a classification of it is only known in dimension 3. Of course another possibility to generalize the previous results is to look at Lagrangian submanifolds of inefinite complex space forms. As a basic ingredient in the positive definite case which is the choice of a canonical frame based on the choice of e_1 as a vector on which a certain function on a compact set attains an absolute maximum breaks down in the indefinite case; new methods need to be developed. Moreover, the above developped techniques can also be used to study some submanifolds in affine differential geometry. The results in these lectures are based on work in progress with F. Dillen, H.Li and X. Wang for Lagrangian submanifolds and with O.Birembaux and M. Djoric for affine differential geometry.

Está conferencia está organizada por el Programa de Postgrado en Matemáticas

Está conferencia está organizada por el Programa de Postgrado en Matemáticas

Conferencia dentro del programa de postgrado Máster de Matemáticas.