# On index expectation curvature for manifolds.

@inproceedings{Knill2020OnIE, title={On index expectation curvature for manifolds.}, author={Oliver Knill}, year={2020} }

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… CONTINUE READING

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