We find the first three most general Minkowski or Hsiung–Minkowski identities relating the total mean curvatures \(H_i\), of degrees \(i=0,1,2,3\), of a closed hypersurface N immersed in a given orientable Riemannian manifold M endowed with any given vector field P. Then we specialize the three identities to the case when P is a position vector field. We further obtain that the classical Minkowski identity is natural to all Riemannian manifolds and, moreover, that a corresponding 1st degree Hsiung–Minkowski identity holds true for all Einstein manifolds. We apply the result to hypersurfaces with constant \(H_1,H_2\).

我们发现前三个最普遍的Minkowski或Hsiung–Minkowski恒等式与浸入给定方向的封闭超曲面N的总平均曲率\（H_i \）的度数\（i = 0,1,2,3 \）有关。黎曼流形中号赋予了任何给定的矢量场P。然后，我们将三个身份专门化为P是位置矢量场的情况。我们进一步获得了经典的闵可夫斯基恒等式对于所有黎曼流形都是自然的，而且，相应的一阶Hsiung-Minkowski恒等式对于所有爱因斯坦流形都成立。我们将结果应用于常数\（H_1，H_2 \）的超曲面。