Let $(X,g)$ be a closed oriented smooth Riemannian manifold of dimension $4$. For each $j$, let $H^j(X;\mathbb R)$ denote de Rham cohomology and let $H_j(X;\mathbb R)$ denote singular homology with real coefficients. Let $[X]\in H_4(X;\mathbb R)$ denote the oriented fundamental class, and write $\langle \alpha,[X]\rangle$ for the evaluation pairing of $\alpha\in H^4(X;\mathbb R)$ on $[X]$. Let $\operatorname{vol}_g\in\Omega^4(X)$ be the Riemannian volume form determined by $g$ and the orientation, and let $d\operatorname{vol}_g$ denote the associated Riemannian volume measure. Let $\nabla$ be the Levi-Civita connection on the tangent bundle $TX\to X$, and let $\Omega\in \Omega^2(X;\operatorname{End}(TX))$ be the curvature form of $\nabla$. Define the first Pontryagin form of $\nabla$ by the Chern-Weil convention

\begin{align*} p_1(\Omega):=-\frac{1}{8\pi^2}\operatorname{tr}(\Omega\wedge\Omega)\in \Omega^4(X), \end{align*}

so that its de Rham cohomology class is the first Pontryagin class $[p_1(\Omega)]=p_1(TX)\in H^4(X;\mathbb R)$. Let $f_\Omega:X\to\mathbb R$ be the unique smooth function such that $p_1(\Omega)=f_\Omega\operatorname{vol}_g$. Let $Q_X:H^2(X;\mathbb R)\times H^2(X;\mathbb R)\to\mathbb R$ be the cup-product intersection form defined by $Q_X(\alpha,\beta):=\langle \alpha\smile\beta,[X]\rangle$, and let $\sigma(X)$ denote the signature of $Q_X$. Then

\begin{align*} \sigma(X)=\frac{1}{3}\int_X f_\Omega\,d\operatorname{vol}_g=\frac{1}{3}\langle p_1(TX),[X]\rangle. \end{align*}