Topics in Differential Geometry

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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.