基于Hodge星算子与水平–垂直分解的向量丛Thom形式构造

Construction of Thom Forms of Vector Bundle via the Hodge Star Operator and Horizontal-Vertical Decomposition

摘要: 本研究基于Hodge星算子构造给出向量丛Thom形式的另一种几何表示。针对经典Berezin积分方法需将曲率嵌入${\Lambda }^{2}V$ 的过程，我们通过保度量联络诱导的典范水平–垂直分解，由曲率张量定义一个在丛空间上的曲率4-形式$R^E$ 。在此基础上，引入指数映射与Hodge星算子构造Thom形式${\alpha }_1={\left(2\pi \right)}^{-n}exp\left(-{\Vert P\Vert }^2/2\right)\star exp\left(R^E\right)$ ，可以证明其与Berezin积分表示一致。相较于传统方法，该构造通过星算子的几何操作显式关联曲率张量与纤维结构，增强了Thom类表示的几何直观性。进一步地，在平坦向量丛情形下，通过此公式，可以直接给出欧拉类表达式。

Abstract: This study presents an alternative geometric representation of the Thom form for vector bundles based on the construction of the Hodge star operator. In contrast to the classical Berezin integral method, which requires embedding the curvature into the cohomological framework, we utilize the canonical horizontal-vertical decomposition induced by a metric-preserving connection to define a curvature 4-form on the total bundle space directly from the curvature tensor. By introducing an exponential map and the Hodge star operator, we construct a Thom form TT, and it is demonstrated that this construction coincides with the Berezin integral representation. Compared to traditional approaches, our method explicitly relates the curvature tensor to the fiber structure through geometric operations involving the Hodge star operator, thereby enhancing the geometric clarity of the Thom class representation. Furthermore, in the case of flat vector bundles, this formula directly yields an explicit expression for the Euler class.