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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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