On index expectation curvature for manifolds
Abstract
Index expectation curvature K(x) = E[i_f(x)] on a compact Riemannian 2dmanifold M is the expectation of PoincareHopf indices i_f(x) and so satisfies the GaussBonnet relation that the interval of K over M is Euler characteristic X(M). Unlike the GaussBonnetChern integrand, such curvatures are in general nonlocal. We show that for small 2dmanifolds M with boundary embedded in a parallelizable 2dmanifold 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.
 Publication:

arXiv eprints
 Pub Date:
 January 2020
 arXiv:
 arXiv:2001.06925
 Bibcode:
 2020arXiv200106925K
 Keywords:

 Mathematics  Differential Geometry;
 57M15;
 53Axx;
 53Cxx
 EPrint:
 13 pages figures