Let $m\ge 0$, and let $(M,g)$ be a closed oriented smooth Riemannian manifold of dimension $2m$. Let $\operatorname{vol}_g\in\Omega^{2m}(M)$ be the Riemannian volume form determined by $g$ and the orientation, and let $d\operatorname{vol}_g$ be the associated Riemannian volume measure. Let $\nabla^{TM}$ be the Levi-Civita connection on the oriented Euclidean vector bundle $(TM,g)$, and let $e(\nabla^{TM})\in\Omega^{2m}(M)$ denote the Euler form defined by the normalized Pfaffian Chern-Weil convention compatible with the orientation of $TM$. Let $E_g:M\to\mathbb R$ be the unique smooth function satisfying $e(\nabla^{TM})=E_g\operatorname{vol}_g$. Then $e(\nabla^{TM})$ is closed, its de Rham cohomology class is the image of the integral Euler class $e(TM)\in H^{2m}(M;\mathbb Z)$ under the coefficient map to $H^{2m}_{\mathrm{dR}}(M)$, and

\begin{align*} \int_M E_g\,d\operatorname{vol}_g=\chi(M). \end{align*}