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We prove that a Gaussian ensemble of smooth random sections of a real vectorbundle over compact manifold canonically defines a metric on the bundletogether with a connection compatible with it. Additionally, we prove a refinedGauss-Bonnet-Chern theorem stating that if the bundle and the manifold areoriented, then the Euler form of the above connection can be identified, as acurrent, with the expectation of the random current defined by the zero-locusof a random section in the above Gaussian ensemble.

A stochastic Gauss-Bonnet-Chern formula