In the present paper, we first characterize real hypersurfaces in the complex quadric \(Q^{m}\) by giving an inequality in terms of the scalar curvature and the Mean curvature vector field. We also obtain the condition under which this inequality becomes an equality. Further, we develop two extremal inequalities involving the normalized \(\delta \)-Casorati curvatures and the extrinsic generalized normalized \(\delta \)-Casorati curvatures for real hypersurfaces in \(Q^{m}\). Finally, we derive the necessary and sufficient condition for the equality in both cases.

中文翻译：

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复二次曲面中实超曲面的广义归一化$$ \δ$$δ-Casorati曲率的界

在本文中，我们首先通过给出标量曲率和平均曲率向量场的不等式，来表征复二次曲面\（Q ^ {m} \）中的实超曲面。我们还获得了这种不平等成为平等的条件。此外，我们针对\（Q ^ {m} \）中的实际超曲面开发了两个极值不等式，其中包括归一化\（\ delta \）- Casorati曲率和外在广义归一化归一化\（\ delta \）- Casorati曲率。最后，我们得出两种情况下平等的必要和充分条件。