Riemannian metric representatives of the Stiefel-Whitney classes
Department of Mathematics, Courant Institute of Mathematical Sciences（纽约大学柯朗数学科学研究所）
Santiago R Simanca
备注：
This version contains a number of editorial corrections to the original, and includes several changes prompted by comments of a referee that improve significantly the presentation
发表信息：
Differential Geometry and its Applications 102 (2026) pp. 1-11
如果M是一个闭合流形，K是M的光滑三角剖分，whitney证明所有stiefel-whitney类都可以作为对偶单元复形(k')*上的cochains指定，将每个对偶单元赋予1 mod 2的值。我们提供一个任意的riemannian度量g的(M,K)对，并利用whitney的标准来证明所有stiefel-whitney类w_1(M),...,w_n(M)都有相应的代表。w_1(M)的代表由detg_ij决定，g_ij是在每个对偶1单元局部定义的框架中计算的；偶类w_{2k}(M)的代表由局部定义的完全测地定向2k流形的chern-gauss-bonnet密度2k-形式决定，每个对偶2k单元关联到一个带边界的2k流形；奇类w_{2k+1}(M)的代表由局部定义的完全测地定向(2k+1)流形的边界球面的超曲面面积形式决定，每个对偶(2k+1)单元关联到一个带边界的(2k+1)流形。如果(M,J,g)是hermitian的，我们证明由此得到的w_{2k}(M)的度量代表是k-th chern类c_k(M,J)诱导的系数同态的z/2约简，而任何奇次类w_{2k+1}(M)的度量代表在上同调中都是平凡的。
英文摘要：
If $M$ is a closed manifold, and $K$ is a smooth triangulation of $M$, Whitney proved that all of the Stiefel-Whitney classes are specified as cochains on the dual cell complex $(K')^*$ assigning the value $1$ mod $2$ to each dual cell. We provide the pair $(M,K)$ with an arbitrary Riemannian metric $g$, and use Whitney's criteria to show that there are associated representatives of all the Stiefel-Whitney classes $w_1(M), \ldots , w_n(M)$. The representative of $w_1(M)$ is determined by $\det{g_{ij}}$, the $g_{ij}$s computed in a frame that is locally defined at each dual $1$-cell; the representatives of the even classes $w_{2k}(M)$ are determined by the Chern-Gauss-Bonnet density $2k$-form of locally defined totally geodesic oriented $2k$ manifolds with boundary associated to each dual $2k$-cell; and the representatives of the odd classes $w_{2k+1}(M)$ are determined by the hypersurface area form of the boundary sphere of a locally defined totally geodesic oriented $(2k+1)$ manifold with boundary associated to each dual $(2k+1)$-cell. If $(M,J,g)$ is Hermitian, we prove that the metric representative of $w_{2k}(M)$ so obtained is the $\mathbb{Z}/2$ reduction of the $k$-th Chern class $c_k(M,J)$ induced by the coefficient homomorphism, and that the metric representative of any odd degree class $w_{2k+1}(M)$ so obtained is trivial in cohomology.
