论文标题
Riemannian歧管的DEHN型数量
A Dehn type quantity for Riemannian manifolds
论文作者
论文摘要
我们查看功能性y（m）= int_m k（x）dv（x）的compact riemannian 2d-manifolds m，其中k（x）=（2d）！ （d！）^ - 1（4pi）^ - d int_t prod_k = 1^d k_t_2k，t_2k+1（x）dt涉及d截面曲率k_ij（x）的d sim t sim o（2d）的d截面曲率的产物。带有k_d（x）=（d！）^ - 1（4pi）^-d sum_sigma prod_k = 1^d k_sigma（2k-1）的离散版本y_disc（m）y_disc（m），sigma（2k）总和在所有置换率上，所有排列的sigma sigma sigma sigma of 1，..，..，2d。与欧拉（Euler）特征不同的是高斯 - 骨网切尔（Gauss-Bonnet-Chern）是int_m k_gbc dv = x（m），y或y_disc的数量通常依赖于度量。我们对y（m）-x（m）感兴趣，因为如果m具有曲率符号E，则y（m）e^d and y_disc（m）为正，而x（m）e^d> 0仅是猜想。我们在一些具体示例中计算y_disc，例如2d-spheres，4-manifold cp^2，6个歧管SO（4）或8-manifold SU（3）。
We look at the functional Y(M) = int_M K(x) dV(x) for compact Riemannian 2d-manifolds M, where K(x) = (2d)! (d!)^-1 (4pi)^-d int_T prod_k=1^d K_t_2k,t_2k+1(x) dt involves products of d sectional curvatures K_ij(x) averaged over the space T sim O(2d) of all orthonormal frames t=(t_1, ... ,t_2d). A discrete version Y_disc(M) with K_d(x) = (d!)^-1 (4pi)^-d sum_sigma prod_k=1^d K_sigma(2k-1),sigma(2k) sums over all permutations sigma of 1,..,2d. Unlike Euler characteristic which by Gauss-Bonnet-Chern is int_M K_GBC dV=X(M), the quantities Y or Y_disc are in general metric dependent. We are interested in Y(M)-X(M) because if M has curvature sign e, then Y(M) e^d and Y_disc(M) are positive while X(M) e^d>0 is only conjectured. We compute Y_disc in a few concrete examples like 2d-spheres, the 4-manifold CP^2, the 6 manifold SO(4) or the 8-manifold SU(3).
