Let $M$ be a compact connected orientable Riemannian $n$-manifold with boundary $\partial M\ne\emptyset$. Since $H^n(M,\mathbb R)=0$, the connecting morphism $\delta: H^{n-1}(\partial M,\mathbb R)\to H^n(M,\partial M,\mathbb R)$ of the exact sequence of the pair $(M,\partial M)$ is an epimorphism, implying that there is an $(n-1)$-form $\alpha$ on $\partial M$ such that volume form of $M$ is $\delta \alpha$. Then $\operatorname{vol}(M)=\delta \alpha([M])=\alpha([\partial M])$.

Let $M$ be hyperbolic with a geodesic boundary now. Did someone figure out some nice general formula for $\alpha$ in that case? $\alpha$ cannot be the volume form on $\partial M$, at least for $n=2$, since the area of $M$ in that case is a multiple of $\pi$, while the length of $\partial M$ can be arbitrary. In fact, for $n=2$ this is the subject of Gauss-Bonnet theorem, so I guess I am asking for a generalization of it to hyperbolic manifolds with geodesic boundary for $n>2.$

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