\(M^n(k)\) denotes an \(n\)-dimensional simply connected Riemannian manifold, of constant sectional curvature \(k\), \(S\) a compact immersed hypersurface of \(M^n(k)\), \(Q\) a compact domain in \(M^n(k)\) with smooth boundary, \(K\) the total curvature of \(S\), \(K_i\) the \(i\)th mean curvature of \(S\), \(\chi\) the Euler characteristic of \(S\). \({\mathcal L}_{n-2}\) the space of \((n- 2)\)-dimensional totally geodesic submanifolds of \(M^n(k)\), \({\mathcal L}_{n-2}\) is a homogeneous space of the isometry group of \(M^n(k)\), with a unique invariant measure \(dL\).
The main result is theorem 1. If \(Q\subset M^n(k)\) is a compact domain with smooth boundary, then \[ \int_{\partial Q} K(x)\,dx= \text{vol}(\mathbb{S}^{n-1})\chi(Q)- k{2(n- 1)\over \text{vol}(\mathbb{S}^{n-2})} \int_{{\mathcal L}_{n-2}}\chi(L\cap Q)\,dL. \] If \(S\) is a compact immersed hypersurface and \(n\) is odd, then \[ \int_S K(x)\,dx= {\text{vol}(\mathbb{S}^{n-1})\over 2} \chi(S)- k{n-1\over \text{vol}(\mathbb{S}^{n-2})} \int_{{\mathcal L}_{n-2}}\chi(L\cap Q)\,dL. \] From this theorem one can obtain \[ \int_S K(x)\,dx= \text{vol}(\mathbb{S}^{n-1})\chi(Q)- \sum_i K_{n-2i-1}(x)\,dx, \]
\[ \int_{\partial Q} K(x)\,dx= \text{vol}(\mathbb{S}^{n-1})\chi(Q)- \sum_i c_k k^i\int_{\partial Q} K_{n-2i-1}(x)\,dx- ck^{n/2} V. \]