Riemannian metric representatives of the Stiefel-Whitney classes

Abstract

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.