[Paper Review] On index expectation curvature for manifolds

Oliver Knill|arXiv (Cornell University)|Jan 20, 2020

Geometric Analysis and Curvature Flows8 references4 citations

TL;DR

This paper introduces a non-local index expectation curvature $ K(x) = \mathbb{E}[i_f(x)] $ on compact Riemannian $ 2d $-manifolds, constructed as a product of sectional index expectation curvatures $ K_k(x) = \mathbb{E}[i_k(x)] $, which satisfy $ \int_M K(x)\,dV(x) = \chi(M) $ and inherit the sign $ e^d $ of the manifold's sectional curvature $ e $. The method relies on integral geometry and Morse functions on small or product manifolds, offering a novel, non-local approach to the Hopf sign conjecture.

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.

Motivation & Objective

• To revive the algebraic Hopf conjecture by constructing curvature functions with sign matching the manifold's sectional curvature.
• To establish a non-local curvature function $ K(x) $ on $ 2d $-manifolds that satisfies Gauss-Bonnet and inherits the curvature sign $ e^d $.
• To demonstrate that such curvature functions can be constructed globally on product manifolds or locally on small boundaryed submanifolds of parallelizable ambient spaces.
• To show that the curvature function $ K(x) $ arises as a product of independent, uncorrelated sectional index expectations $ K_k(x) $, ensuring sign propagation.
• To provide a framework for constructing curvature functions that are absolutely continuous and piecewise smooth, even when global smooth constructions fail.

Proposed method

• Define index expectation curvature $ K(x) = \mathbb{E}[i_f(x)] $ as the expected Poincaré-Hopf index over a probability space of Morse functions.
• Construct sectional index expectation curvatures $ K_k(x) = \mathbb{E}[i_k(x)] $ by averaging indices over planes parallel to coordinate 2-planes in a fixed frame.
• Use independence of sectional indices $ i_k(x) $ across different planes to derive the product formula $ K(x) = \prod_{k=1}^d K_k(x) $ via decorrelation of expectations.
• Restrict to small manifolds with boundary or product manifolds $ M = M_1 \times \cdots \times M_d $, where global frame splitting enables consistent curvature construction.
• Utilize integral geometry and parametrized surfaces $ \Sigma: z_k \mapsto f(\dots, z_k, \dots) $ to define curvature averages, ensuring sign consistency under definite curvature assumptions.
• Apply the product lemma: for independent random variables $ i_k(x) $, $ \mathbb{E}[\prod i_k(x)] = \prod \mathbb{E}[i_k(x)] $, validating the product structure in the interior of $ M $.

Experimental results

Research questions

• RQ1 Can a non-local curvature function $ K(x) $ be constructed on a compact $ 2d $-manifold such that $ \int_M K(x)\,dV(x) = \chi(M) $ and $ K(x) $ has sign $ e^d $?
• RQ2 Does the product structure $ K(x) = \prod_{k=1}^d K_k(x) $ of sectional index expectation curvatures preserve the sign $ e^d $ when the sectional curvatures have sign $ e $?
• RQ3 Is it possible to define such curvature functions globally on general $ 2d $-manifolds, or are local or product structures necessary due to topological obstructions?
• RQ4 How do boundary contributions affect the curvature function $ K(x) $, and can they be consistently incorporated in the index expectation framework?
• RQ5 Can the algebraic Hopf conjecture be revived by constructing curvature functions that are absolutely continuous and piecewise smooth with correct sign?

Key findings

• The index expectation curvature $ K(x) = \mathbb{E}[i_f(x)] $ satisfies the Gauss-Bonnet relation $ \int_M K(x)\,dV(x) = \chi(M) $ on compact $ 2d $-manifolds.
• For small $ 2d $-manifolds with boundary embedded in a parallelizable $ 2d $-manifold $ N $ with definite curvature sign $ e $, the curvature function $ K(x) $ has sign $ e^d $ in the interior.
• The curvature $ K(x) $ is constructed as a product $ \prod_{k=1}^d K_k(x) $, where each $ K_k(x) = \mathbb{E}[i_k(x)] $ is the expectation of a sectional index, and the independence of $ i_k(x) $ ensures $ \mathbb{E}[\prod i_k(x)] = \prod \mathbb{E}[i_k(x)] $.
• The construction fails globally on non-parallelizable manifolds due to the absence of global orthonormal frame sections, necessitating local or product decompositions.
• In the product case $ M = M_1 \times \cdots \times M_d $, where each $ M_k $ is a 2-manifold, the curvature function $ K(x) $ can be globally defined and is smooth with sign $ e^d $.
• Boundary contributions are non-zero and independent of the frame bundle choice, with curvature mass concentrated on vertices or edges in polyhedral decompositions.