权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

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伯杰列表中出现的不同类型的几何有关。这个列表分类了非对称单连通黎曼流形的所有可能的完整群。这些几何中有几个是自动的Ricci平坦的，并且拥有平行的旋量场。这两个性质使它们对物理学家也非常有吸引力，它们在物理学中作为“内部空间”出现在高维超对称理论的紧化中。更一般地说，物理学家使用具有几何结构的内部空间，这些空间具有所谓的“特征”联系。第一部分是关于Kähler几何和超Kähler几何的某些推广，第二部分研究与Berger列表中的Ricci-平坦几何相关的不同几何流方程。第一部分更精确地处理SKT-结构，这是具有特征连接的Kähler结构的推广，以及复辛结构，这是超Kähler结构的推广。在这两种情况下，我们的目标是与其他数学家合作，在不同类的幂零和可解李群的左不变上下文中对这些结构进行分类。请注意，在SKT的情况下，我们将使用剪切构造进行分类，这是我和我的合作者在项目的这一部分中共同开发的一种构造。项目的第二部分是关于旋量流，修改的拉普拉斯协流以及希钦流和群收缩之间的相互作用。前两个流的临界点分别是任意的或七维的Ricci平坦的黎曼流形（附加数据），它们来自Berger的列表。我们的目的是研究这些相对较新的流动在一个均匀的设置和其他“对称”的情况下，以获得更好地了解这些流量一般的例子和属性。希钦流是一个几何流在六个维度产生七维Ricci平坦黎曼流形与holonomy在G2（一个“例外”的情况下，在伯杰的名单）。通过所谓的群压缩，物理学家从S^3 × S^3上希钦流的左不变解构造出该流在六维李群上的左不变解。我们现在的目标是详细地理解群收缩和希钦流之间的这种相互作用，并希望在S^3 × S^3和其他李群上系统地研究这种相互作用。