Topics in Differential Geometry

虽然接触几何是微分几何和经典力学中的经典学科，但在过去的25年里，它也是一个从多个不同角度进行深入研究的学科。该提案的PI将重点放在接触结构的微分几何方面。传统上，接触结构被认为是辛结构的奇维类比。在过去的五年里，对辛结构的研究扩展到了广义复结构的范畴。在这个方案中，PI将发展相应的关于奇维流形上的接触1-形式的理论，重点是可积性和形变。虽然所提出的理论的出发点与广义复杂结构的出发点相似，但相关的可积性概念和随后的形变理论预计会有本质的不同。该项目的完成将使接触结构能够以非平凡和可控的方式变形，推广了Sasakian几何中的变形理论，并发展了奇维空间上的弱镜对称理论。它的现代需求源于理论物理。在数学上，这个项目关注的是一个人管理和操纵由奇数维空间定义的方程组的解的能力。这些方程是由空间的几何或物理性质决定的。这个项目是建立在过去五年中关于偶数维空间的相应理论的发展基础上的。然而，有关空间的维度的奇偶性的不同在几何及其相关方程的性质上造成了显著的差异。因此，这个项目将开发新的概念和新的工具来管理差异。

While contact geometry is a classical subject in differential geometry and classical mechanics, it is also a subject of intensive investigation from many different perspectives in the past twenty five years. The PI of this proposal will focus on the differential geometric aspects of contact structures. Traditionally, contact structures are recognized as an odd-dimensional analogue of symplectic structures. In the past five years, investigation on symplectic structures is extended to the category of generalized complex structures. In this proposal, the PI will develop a corresponding theory for contact 1-forms on odd-dimensional manifolds, with a focus on integrability and deformation. While the starting point of proposed theory is similar to that of generalized complex structures, the related concept on integrability and the subsequent deformation theory are expected to the substantially different. A completion of this project will enable contact structures to deform in a non-trivial and controlled manner, extend a deformation theory on Sasakian geometry and develop a weak mirror symmetry theory on odd-dimensional spaces.The origin of the topics for investigation in this project could be traced to classical mechanics. Its modern needs are due to theoretical physics. Mathematically, this project is concerned with one's ability to manage and manipulate the solutions of a system of equations defined an odd-dimensional spaces. The equations are determined by the geometric or physical nature of the spaces. This project is built upon the development of a corresponding theory on even-dimensional spaces in the past five years. However, the difference in the parity of the dimension of the concerned spaces poses striking difference in the nature of the geometry and its related equations. Therefore, this project will development new concepts and new tools to manage the difference.