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

与 Ricci 平黎曼流形相关的（超）克勒几何和几何流的推广

基本信息

批准号：
405980393
负责人：
Dr. Marco Karl Freibert
金额：
--
依托单位：
Mathematisches Seminar
依托单位国家：
德国
项目类别：
Research Fellowships
财政年份：
2018
资助国家：
德国
起止时间：
2017-12-31 至 2019-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/405980393?language=en
关键词：
Generalizations hyper hler geometry geometric

项目摘要

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.

我的研究项目与所谓的Berger列表中发生的不同几何形状有关。此列表将无与伦比的简单歧管歧管分类所有可能的自动组。这些几何形状中的几个是自动的，并且潜在的平行旋转型场。这两种特性使它们对物理学家也非常有吸引力，并且它们在物理学中作为“内部空间”在高维超对称理论的压缩中。更普遍地，物理学家使用具有所谓“特征”连接的几何结构的内部空间。我的研究项目可以大致分为两个部分。尽管第一部分是在Kähler和Hyperkähler几何形状的某些概括上，而第二部分则检查与Berger列表中与Ricci-Flat几何形状相关的不同的几次流动方程式，第一部分更与SKT结构更加准确地涉及SKT结构，这些结构是Kähler结构的概括性结构的概括性结构，并且是综合结构的特征性结构，并且是综合体的构造，并且是综合体系的构造。在这两种情况下，目标都是与其他数学家合作，在不同类别的nilpotent and-able可解决的谎言群体上对这些结构进行分类。请注意，在SKT案例中，我们将使用剪切构建来进行分类，这是一种与我在项目的这一部分中的合作者一起开发的结构。项目的第二部分在于旋转型流，修改后的Laplacian Coflow和Hitchin Flow和Hitchin Flow和组收缩之间的相互作用。同样，所有这些子项目都是与伦敦和欧洲其他地方的不同数学家的合作。前两个流的关键要点分别是任意或七维的ricci-flat riemannian歧管（还有贝尔格列表的其他数据）。我们的目的是在同质环境和其他“对称”案例中研究这些相对新流的实例和特性，以便更好地了解这些流量。希钦流量是六个维度的几何流程，在六个维度中产生七维ricci-flat riemannian riemannian riemannian practords in G2中的自由主义者（一个例外情况）。通过所谓的群体收缩，物理学家在S^3 \ times s^3上的该流量的左右溶液中构建了Hitchin流的剩余解决方案。我们现在的目标是在恰好提到的情况下详细了解小组收缩与Hitchin流量之间的这种相互作用，并希望系统地研究S^3 \ times S^3和其他谎言组的这种相互作用。