在本計劃中，我們研究CR幾何包括1. 海森堡群中子流形的基本定理：我們發現二維海森堡群中二維閉的Legendrian曲面的拓撲不變量。我們想研究二維海森堡群中Legendrian曲面的contact不變量。2.CR幾何中的Q-曲率：令嵌入N維複平面的（2n + 1）維流形M是由定義函數u定義的超曲面。我們想研究如何通過定義函數u來表示psudohermitian流形M的幾何量。如果connection form具有定義函數u的良好表示，那麼我們希望我們可以刻劃contact formθ，使得Q曲率等於0。3.在n維複平面中刻劃球：設M是嚴格擬凸域的邊界。通常，M的定義函數u確定所有這樣的CR流形M。因此，我們希望通過研究定義函數來研究M，並且想要通過定義函數來獲得一些結果。4. Conformal愛因斯坦空間和Bach tensor：由於陳省身的曲率張量是類似於Wely張量，我們想研究由陳省身的pseudo-conformal曲率張量所定義出來的W functional，找到這個functional和拓撲量之間的關係。另外，我們想研究在psudohermitian manifolds上類似於Bach張量的張量。 ;In this project, we study the CR geometry including: 1. The fundamental theorem of submanifolds in Heisenberg group: We find the topological invariant for the 2-dimensional closed oriented Legendrian surfaces in 2-dimensional Heisenberg group H₂. We would like to study the contact invariants of the Legendrian surface of H₂. 2. Q-Curvature in CR Geometry: Let M be (2n+1)-dimensional hypersurface in N-dimensional complex plane and M defined by a defining function u. We would like to study how to represent the geometry quantities of the psudohermitian manifold M by the defining function u. If the connection form has a good representation of defining function u , then we hope that we can characterize the contact form θ such that Q-curvature vanishes. 3. Characterizing balls in Cⁿ: Let M be the boundary of a strictly pseudoconvex domain. In general, a defining function u of M determins all about the CR manifold M (or pseudohermitian manifold). Therefore, we would like to study M by study defining functions, and like to get some results of which can be described by using the defining functions. 4. Conformal Einstein spaces and Bach tensor: In psudohermitian manifolds, since Chern's pseudo-conformal curvature tensor is analogous to the Weyl tensor, we would like to study the W functional of Chern's pseudo-conformal curvature tensor and find the relationship between this functional and the topological quantities. Moreover, we would like to study the tensor in psudohermitian manifolds which is analogous to the Bach tensor.
