Homogeneous geometry in low dimension: constructions and classifications

The Project aims to unite forces of the Norwegian and German research groups and to focus on several important research problems of common interest described below. 1. Classification of homogeneous strictly nearly pseudo-Kahler spaces in 6 dimensions. Both groups have an active interest in this problem and some partial results. Further results on inhomogenous strictly nearly pseudo-Kahler spaces with abundant symmetry are plausible. 2. Geometric structures on naturally reductive spaces. The group in Marburg has considerable success in classification of naturally reductive homogeneous spaces. We will investigate the underlying geometric structures in dimension 7 and 8, in particular, G2 and Spin7-structures. Affine connections with skew torsion will be also studied in relation to integrable systems. 3. Maximal symmetry of Gray-Hervella classes. This is a new problem of definite interest to both groups. Not much is known in this problem notwithstanding 35 years of active study of these 16 classes of manifolds in dimension 6. Both sides have own approaches, which are complementary, and so a joint study should be very prolific. 4. Spinorial properties of connections with non-vanishing torsion. Solutions of the spinorial field equations are crucial for applications to theoretical physics. Only few examples are known so far. By computer experiments and real Lie theory, we will carry out a systematic search for new solutions and describe them. 5. Representation of highly symmetric structures on homogeneous spaces. This problem of representing the space with a geometric structure by the model G/H with various (G,H) can be traced back to the works of Sophus Lie, yet not much is known in general. We will try to find new approaches basing on the previous investigations and results of the Tromsø group. The two groups have had preliminary contacts and meetings, which give a definite hope to construct a fruitful dialog and lay a foundation for future collaborations.