On index expectation curvature for manifolds
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.
- Publication:
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arXiv e-prints
- Pub Date:
- January 2020
- DOI:
- 10.48550/arXiv.2001.06925
- arXiv:
- arXiv:2001.06925
- Bibcode:
- 2020arXiv200106925K
- Keywords:
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- Mathematics - Differential Geometry;
- 57M15;
- 53Axx;
- 53Cxx
- E-Print:
- 13 pages figures