We obtain a sharp estimate to the norm of the traceless second fundamental form of complete hypersurfaces with constant scalar curvature immersed into a locally symmetric Riemannian manifold obeying standard curvature constraints (which includes, in particular, the Riemannian space forms with constant sectional curvature). When the equality holds, we prove that these hypersurfaces must be isoparametric with two distinct principal curvatures. Our approach involves a suitable Okumura type inequality which was introduced by Meléndez (Bull Braz Math Soc 45:385–404, 2014) , corresponding to a weaker hypothesis when compared with to the assumption that these hypersurfaces have a priori at most two distinct principal curvatures.

我们对具有恒定标量曲率的完整超曲面的无痕第二基本形式的范数进行了精确的估计，这些标量曲率沉浸在遵循标准曲率约束的局部对称黎曼流形中（尤其包括具有恒定截面曲率的黎曼空间形式）。当等式成立时，我们证明这些超曲面必须是等参的，具有两个不同的主曲率。我们的方法涉及由Meléndez（Bull Braz Math Soc 45：385–404，2014）引入的合适的Okumura型不等式，与假设这些超曲面最多具有两个先验主曲率的先验假设相比，该假设不成立。 。