The Euler Class from a General Connection, Relative to a Metric

We extend the well-known formula for the Euler class of a real oriented even-dimensional vector bundle in terms of the curvature of a metric connection to the case of a general linear connection provided a metric is present. We rewrite the classical Gauss-Bonnet theorem in dimension two in light of this formula. We also discuss a potential application to a conjecture of Chern, and make a brief digression to discuss $m$-quasi-Einstein manifolds.