论文标题
在索引期望曲率上
On index expectation curvature for manifolds
论文作者
论文摘要
在紧凑的Riemannian 2d-manifold m上，索引期望曲率k（x）= E [i_f（x）]是对Poincare-Hopf指数I_F（X）的期望，因此可以满足Guss-Bonnet的关系，即M的间隔在m上超过m是Euler特征x（m）。与高斯 - 骨网络整合体不同，这种曲率通常是非本地的。我们表明，对于具有嵌入边界的小型2D manifolds m，在确定的截面曲率符号E中可行的2D-manifold n中，存在确定符号E^d的索引期望k（x）。函数k（x）被构造为截面期望值曲率的乘积，平均k_k（x）= e [i_k（x）]莫尔斯函数的概率空间f，i_f（x）是i_k（x）的产物I_K（x）的产物，I_K是独立的，并且不相关。
Index expectation curvature K(x) = E[i_f(x)] on a compact Riemannian 2d-manifold M is the expectation of Poincare-Hopf indices i_f(x) and so satisfies the Gauss-Bonnet relation that the interval of K over M is Euler characteristic X(M). Unlike the Gauss-Bonnet-Chern integrand, such curvatures are in general non-local. We show that for small 2d-manifolds M with boundary embedded in a parallelizable 2d-manifold N of definite sectional curvature sign e, an index expectation K(x) with definite sign e^d exists. The function K(x) is constructed as a product of sectional index expectation curvature averages K_k(x) = E[i_k(x)] of a probability space of Morse functions f for which i_f(x) is the product of i_k(x), where the i_k are independent and so uncorrelated.
