Generalizations of (hyper-)Kähler geometry and geometric flows related to Ricci-flat Riemannianmanifolds

点击翻译按钮获取中文摘要

My research project is related to the different kinds of geometries occuring in the so-called Berger list. This list classifies all possible holonomy groups of non-symmetric simply-connected Riemannian manifolds. Several of these geometries are automatically Ricci-flat and possess a parallel spinor field. These two properties make them also very attractive for physicists and they occur in physics as "internal spaces" in compactifications of higher-dimensional supersymmetric theories. More generally, physicists use internal spaces with geometric structures which possess a so-called "characteristic" connection.My research project can be divided roughly into two parts. Whereas the first part is on certain generalizations of Kähler and hyperkähler geometry, the second part examines different geometric flow equations related to the Ricci-flat geometries from Berger's list.The first part deals more exactly with SKT-structures, which are generalizations of Kähler structures which possess a characteristic connections, and with complex-symplectic structures, which are generalizations of hyperkähler structures. In both cases, the goal is to classify, in cooperation with other mathematicians, these structures in a left-invariant context on different classes of nilpotent and solvable Lie groups. Note that in the SKT case, we will use the shear-construction for the classification, a construction which has been developped before together with my collaborator in this part of the project.The second part of the projects is on the spinor flow, the modified Laplacian coflow and the interplay between the Hitchin flow and group contractions. Also all these subprojects are collaborations with different mathematicians from London and other places in europe.The critical points of the first two flows are arbitrary or seven-dimensional Ricci-flat Riemannian manifolds (with additional data) from Berger's list respectively. Our aim is to study examples and properties of these relatively new flows in a homogeneous setting and other "symmetric" cases in order to gain a better understanding of these flows in general.The Hitchin flow is a geometric flow in six dimensions which produces seven-dimensional Ricci-flat Riemannian manifolds with holonomy in G2 (an "exceptional" case in Berger's list). Via so-called group contractions, physicists constructed left-invariant solutions of the Hitchin flow on six-dimensional Lie groups from left-invariant solutions of that flow on S^3\times S^3. We aim now at understanding this interplay between group contractions and the Hitchin flow in detail in the just mentioned cases and want to study this interplay systematically on S^3\times S^3 and other Lie groups.