Let $(X,g)$ be a closed oriented smooth Riemannian manifold of dimension $4$. Let $\operatorname{vol}_g\in \Omega^4(X)$ be the Riemannian volume form determined by $g$ and the orientation, and let $d\operatorname{vol}_g$ be the associated Riemannian volume measure. Let $\nabla^{LC}$ be the Levi-Civita connection on $TX$, let $R^g$ be its Riemann curvature tensor, and let $\Omega\in \Omega^2(X;\mathfrak{so}(TX,g))$ be its curvature form. Define the Euler form $e(\Omega)\in \Omega^4(X)$ by the Pfaffian Chern-Weil normalization determined by the given orientation, and let $E_g:X\to\mathbb R$ be the unique smooth function satisfying $e(\Omega)=E_g\operatorname{vol}_g$. Then

\begin{align*} \chi(X)=\int_X E_g(x)\,d\operatorname{vol}_g(x). \end{align*}

Equivalently, for each $p\in X$, let $\mathcal R_p:\Lambda^2T_p^*X\to\Lambda^2T_p^*X$ be the Riemannian curvature operator characterized as follows: for every positively oriented $g_p$-[orthonormal basis](/page/Orthonormal%20Basis) $(e_1,e_2,e_3,e_4)$ of $T_pX$, with [dual basis](/theorems/414) $(e_1^*,e_2^*,e_3^*,e_4^*)$, define $R_{ijkl}(p):=g_p(R^g(e_i,e_j)e_k,e_l)$. If $\alpha,\beta\in\Lambda^2T_p^*X$ have coordinate expansions

\begin{align*} \alpha=\frac{1}{2}\sum_{i,j=1}^{4}\alpha_{ij}e_i^*\wedge e_j^*. \end{align*}

\begin{align*} \beta=\frac{1}{2}\sum_{k,l=1}^{4}\beta_{kl}e_k^*\wedge e_l^*, \end{align*}

then

\begin{align*} (\mathcal R_p(\alpha),\beta)_{\Lambda^2}=\frac{1}{4}\sum_{i,j,k,l=1}^{4}R_{ijkl}(p)\alpha_{ij}\beta_{kl}. \end{align*}

Decompose $\Lambda^2T_p^*X=\Lambda^2_+T_p^*X\oplus\Lambda^2_-T_p^*X$ by the Hodge star determined by $g_p$ and the orientation. Let $W^+_p:\Lambda^2_+T_p^*X\to\Lambda^2_+T_p^*X$ and $W^-_p:\Lambda^2_-T_p^*X\to\Lambda^2_-T_p^*X$ be the diagonal trace-free Weyl curvature blocks of $\mathcal R_p$, let $\operatorname{Ric}_0(p)$ be the trace-free Ricci tensor at $p$, and let $S(p)$ be the scalar curvature at $p$. The pointwise norms $|W^+_p|$ and $|W^-_p|$ are the Hilbert-Schmidt norms of these endomorphisms of $\Lambda^2_+T_p^*X$ and $\Lambda^2_-T_p^*X$, while $|\operatorname{Ric}_0(p)|$ is the tensor norm induced by $g_p$. Then

\begin{align*} \chi(X)=\frac{1}{8\pi^2}\int_X\left(|W^+|^2+|W^-|^2-\frac{1}{2}|\operatorname{Ric}_0|^2+\frac{S^2}{24}\right)(x)\,d\operatorname{vol}_g(x). \end{align*}