论文标题
Stiefel-Whitney班的Riemannian指标代表
Riemannian metric representatives of the Stiefel-Whitney classes
论文作者
论文摘要
如果$ m $是封闭的多种流形，而$ k $是$ m $的平滑三角剖分，惠特尼证明了所有Stiefel-Whitney类都被指定为双电池复合物上的Cochains $（k'）^*$分配值$ 1 $ mod $ 2 $ 2 $ 2 $ 2 $ 2 $。我们为这对$（m，k）$提供了任意的Riemannian Metric $ g $，并使用惠特尼的标准表明所有Stiefel-Whitney类的相关代表$ W_1（m），\ ldots，w_n（m）$。 $ w_1（m）$的代表由$ \ det {g_ {ij}} $，$ g_ {ij} $ s计算的$ g_ {ij} $ s以本地定义的每个双$ 1 $ -cell;均匀类的代表$ w_ {2k}（m）$由Chern-Gauss-Bonnet密度$ 2K $ 2K $ - 本地定义的全部定义完全测量的$ 2K $歧管，与每个双$ 2K $ -CELL相关的边界；奇数类的代表$ w_ {2k+1}（m）$取决于本地定义的边界球的超出面区域形式完全定向的$（2K+1）$歧管，与每个dual dual $（2k+1）$ - cell的边界相关。如果$（M，J，G）$是Hermitian，我们证明$ W_ {2K}（M）$的指标代表如此获得的是$ \ Mathbb {Z}/2 $减少$ k $ -tth Chern Chern类$ c_k（m，j）$ sugy the coffical coldicative nock an $ w_ {2k+1}（m）$因此在同时学中是微不足道的。
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.
