The Gauss-Bonnet-Chern theorem: A probabilistic perspective
Copyright policy )The Gauss-Bonnet-Chern theorem: A probabilistic perspective
We prove that the Euler form of a metric connection on real oriented vector bundle $E$ over a compact oriented manifold $M$ can be identified, as a current, with the expectation of the random current defined by the zero-locus of a certain random section of the bundle. We also explain how to reconstruct probabilistically the metric and the connection on $E$ from the statistics of random sections of $E$.
35 pages, references added, fixed typos, to appear Trans. Amer. Math. Soc
- University of Notre Dame United States
Mathematics - Differential Geometry, Mathematics - Analysis of PDEs, FOS: Mathematics, Euler form, heat kernel asymptotics, Integral geometry, wave kernel asymptotics, currents, random sections, Asymptotic distributions of eigenvalues in context of PDEs, Gauss-Bonnet-Chern theorem, Probability (math.PR), Heat and other parabolic equation methods for PDEs on manifolds, connections, Global Riemannian geometry, including pinching, 35P20, 53C65, 58J35, 58J40, 58J50, 60D05, Gaussian measures, Differential Geometry (math.DG), curvature, Pseudodifferential and Fourier integral operators on manifolds, Geometric probability and stochastic geometry, Laplacian, Mathematics - Probability, Analysis of PDEs (math.AP)
Mathematics - Differential Geometry, Mathematics - Analysis of PDEs, FOS: Mathematics, Euler form, heat kernel asymptotics, Integral geometry, wave kernel asymptotics, currents, random sections, Asymptotic distributions of eigenvalues in context of PDEs, Gauss-Bonnet-Chern theorem, Probability (math.PR), Heat and other parabolic equation methods for PDEs on manifolds, connections, Global Riemannian geometry, including pinching, 35P20, 53C65, 58J35, 58J40, 58J50, 60D05, Gaussian measures, Differential Geometry (math.DG), curvature, Pseudodifferential and Fourier integral operators on manifolds, Geometric probability and stochastic geometry, Laplacian, Mathematics - Probability, Analysis of PDEs (math.AP)
selected citations These citations are derived from selected sources.This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).5 popularity This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.Average influence This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).Average impulse This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.Average
Citations provided by BIP!Popularity provided by BIP!