本项目主要研究Finsler旋流形及其几何性质与拓扑、代数结构和物理的关系。核心思想是利用Finsler-Bochner技巧及相应几何量证明消灭定理并探讨Finsler旋结构在李代数、复流形和理论物理中的应用。该研究方法具有普适性，其结果有重要的理论价值。通过推广Weitzenböck公式、考察调和旋空间、计算Finsler流形的Chern-Simons作用量的临界形式，可得到Finsler旋量场的强消灭定理、Kähler-Finsler全纯上同调群的同构定理、退化椭圆算子的几何刻画等结果，并初步描述高阶旋结构度规场的Chern-Simons理论。其中所反映出的关键科学问题是Finsler几何量的物理意义及其拓扑影响。项目的研究将拓展Finsler几何的理论框架和应用范围，并为理论物理提供一种更广泛的几何背景。因此，对Finsler几何的研究与发展有着重要的理论和应用价值。

Studying the effects of metrics on the topological properties of a manifold is an essential problem in differential geometry. Although introduced for the first time a century ago, Finsler metrics is still far from mature in applications to topological, algebraic and physical structures on manifolds. In this proposal, we mainly study the Finsler spin manifolds and the relations between its geometric properties and its physical, topological and algebraic structures. The essential method is to use the Finsler-Bochner technique and the relevant concepts to study various vanishing theorems..The research method holds good university, and its results have important theoretical values. The research includes three aspects: (1) Strong vanishing theorem on the Finsler spin manifold and its applications to Lie algebraic structures of Finsler manifolds; (2) Relations between the spinor and the holomorphic cohomology groups, as well as between the degenerated Dirac operators and canonical line bundle, on the complex Finsler manifolds; (3) Finsler Chern-Simons action with its Euler-Lagrange equation, and the Finsler Kapustin-Witten Weitzenböck formula..The above three aspects describe the geometric character of the degenerated elliptic operator and describe elementally the Chern-Simons theory of the three dimensional SL(N,R)×SL(N,R) under the Finsler metrics (i.e., the applications of the higher order spinor gauge fields). This research will extend the applications of Finsler geometry on mathematical physics and theoretical physics. Furthermore, it will also provide a new geometric explanation to some issues of modern mathematics. The study of this project would be quite meaningful for the research and development of Finsler geometry and its applications.