The Gauss-Bonnet-Chern theorem: A probabilistic perspective

Nicolaescu, Liviu; Savale, Nikhil

doi:10.1090/tran/6895

The Gauss-Bonnet-Chern theorem: A probabilistic perspective
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by Liviu I. Nicolaescu and Nikhil Savale PDF

Trans. Amer. Math. Soc. 369 (2017), 2951-2986 Request permission

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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