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Transactions of the American Mathematical Society

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The Gauss-Bonnet-Chern theorem: A probabilistic perspective


Authors: Liviu I. Nicolaescu and Nikhil Savale
Journal: Trans. Amer. Math. Soc. 369 (2017), 2951-2986
MSC (2010): Primary 35P20, 53C65, 58J35, 58J40, 58J50, 60D05
DOI: https://doi.org/10.1090/tran/6895
Published electronically: November 28, 2016
MathSciNet review: 3592534
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Abstract | References | Similar Articles | Additional Information

Abstract: We prove that the Euler form of a metric connection on a 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$.


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Additional Information

Liviu I. Nicolaescu
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556-4618
Email: nicolaescu.1@nd.edu

Nikhil Savale
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556-4618
Email: nsavale@nd.edu

DOI: https://doi.org/10.1090/tran/6895
Keywords: Gauss-Bonnet-Chern theorem, currents, random sections, Gaussian measures, connections, curvature, Euler form, Laplacian, wave kernel asymptotics, heat kernel asymptotics
Received by editor(s): November 21, 2014
Received by editor(s) in revised form: September 15, 2015, and December 16, 2015
Published electronically: November 28, 2016
Article copyright: © Copyright 2016 American Mathematical Society

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