• Kyungpook Mathematical Journal
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• v.62 no.1
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• pp.133-166
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• 2022

Let M be a semi-invariant submanifold with almost contact metric structure (𝜙, 𝜉, 𝜂, g) of codimension 3 in a complex space form M_n+1(c) for c ≠ 0. We denote by S and R_𝜉 be the Ricci tensor of M and the structure Jacobi operator in the direction of the structure vector 𝜉, respectively. Suppose that the third fundamental form t satisfies dt(X, Y) = 2𝜃g(𝜙X, Y) for a certain scalar 𝜃 ≠ 2c and any vector fields X and Y on M. In this paper, we prove that if it satisfies R_𝜉𝜙 = 𝜙R_𝜉 and at the same time S𝜉 = g(S𝜉, 𝜉)𝜉, then M is a real hypersurface in M_n(c) (⊂ M_n+1(c)) provided that $\bar{r}-2(n-1)c{\leq}0$, where $\bar{r}$ denotes the scalar curvature of M.

• Bulletin of the Korean Mathematical Society
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• v.52 no.2
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• pp.557-570
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• 2015

Let $\mathbb{G}$ be a von Neumann algebraic locally compact quantum group, in the sense of Kustermans and Vaes. In this paper, as a consequence of a notion of amenability for actions of Lau algebras, we show that $\hat{\mathbb{G}}$, the dual of $\mathbb{G}$, is co-amenable if and only if there is a state $m{\in}L^{\infty}(\hat{\mathbb{G}})^*$ which is invariant under a left module action of $L^1(\mathbb{G})$ on $L^{\infty}(\hat{\mathbb{G}})^*$. This is the quantum group version of a result by Stokke [17]. We also characterize amenable action of Lau algebras by several properties such as fixed point property. This yields in particular, a fixed point characterization of amenable groups and H-amenable representation of groups.