On index expectation curvature for manifolds

2020/01/20 by Oliver Knill, Knill, Oliver · 4 citations
Computer Science · Mathematics · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Topological and Geometric Data Analysis #math.DG #msc:53Axx #msc:53Cxx #msc:57M15

paper · pdf · doi:10.48550/arxiv.2001.06925

13 pages figures

arxiv created 2020/01/20 · arxiv updated 2020/01/22

Abstract

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.

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