Riemann-Finsler和Lagrange Gerbes与Aiyah-Singer定理
Riemann-Finsler and Lagrange Gerbes and the Aiyah--Singer Theorems
摘要：
本文将对流形和具有不可积分布的向量丛空间（简称非完整空间）自然地导出非完整格贝。这种gerbes的一个重要例子是与定义非线性连接（N-连接）结构的分布有关。他们在几何上统一和发展了黎曼-嘉当流形和拉格朗日-芬斯勒空间的概念。非完整空间中自旋结构存在的障碍是第二类Stiefel-Whitney类，由与$\mathbb{Z} /2$ gerbe相关的上循环定义，称为非完整自旋gerbe。非完整格贝正则地被赋予了N-联络、Sasaki型度量、正则线性联络联络和几乎复结构（对于奇维空间）。非完整自旋结构和格贝的研究具有几何和物理两方面的应用。我们的目的是证明Atiyah-Singer定理，这样的非完整空间。
英文摘要：
In this paper, nonholonomic gerbes will be naturally derived for manifolds and vector bundle spaces provided with nonintegrable distributions (in brief, nonholonomic spaces). An important example of such gerbes is related to distributions defining nonlinear connection (N-connection) structures. They geometrically unify and develop the concepts of Riemann-Cartan manifolds and Lagrange-Finsler spaces. The obstruction to the existence of a spin structure on nonholonomic spaces is just the second Stiefel-Whitney class, defined by the cocycle associated to a $\mathbb{Z} /2$ gerbe, which is called the nonholonomic spin gerbe. The nonholonomic gerbes are canonically endowed with N-connection, Sasaki type metric, canonical linear connection connection and (for odd dimension spaces) almost complex structures. The study of nonholonomic spin structures and gerbes have both geometric and physical applications. Our aim is to prove the Atiyah--Singer theorems for such nonholonomic spaces.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
