## Characterization of totally $$\eta$$-umbilic real hypersurfaces in nonflat complex space forms by some inequality.(English)Zbl 1074.53011

Let $$\widetilde{M}_n(c)$$ be an $$n$$-dimensional nonflat complex space form of constant holomorphic curvature $$c$$ which is either a complex projective space $$\mathbb{C}P^n(c)$$ or a complex hyperbolic space $$\mathbb{C}H^n(c)$$. It is known that there is no totally umbilic real hypersurface in $$\widetilde{M}_n(c)$$, but there exist totally $$\eta$$-umbilic real hypersurfaces. A real hypersurface $$M$$ of $$\widetilde{M}_n(c)$$ is called totally $$\eta$$-umbilic if its shape operator $$A$$ is of the form $$AX=\alpha X$$ for each vector $$X$$ on $$M$$ which is orthogonal to the characteristic vector $$\xi$$ of $$M$$, where $$\alpha$$ is a smooth function on $$M$$. The main result of the paper under review is the following:
Theorem. Let $$M$$ be a real hypersurface in a nonflat complex space form $$\widetilde{M}_n(c)$$ ($$n \geq 2$$). Then the following inequality holds: $$(\operatorname{trace} A -\langle A\xi,\xi\rangle)^2 \leq 2(n-1)(\operatorname{trace} A^2 -\| A\xi\| ^2)$$, where $$A$$ is the shape operator of $$M$$ in the ambient space $$\widetilde{M}_n(c)$$. Moreover, the equality holds on $$M$$ if and only if $$M$$ is totally $$\eta$$-umbilic in $$\widetilde{M}_n(c)$$.

### MSC:

 53B25 Local submanifolds 53C40 Global submanifolds 53C55 Global differential geometry of Hermitian and Kählerian manifolds
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### References:

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