### Hypersurfaces with constant $k$-th mean curvature in a Lorentzian space form

In this paper, we study $n(n\ge 3)$-dimensional complete connected and oriented space-like hypersurfaces ${M}^{n}$ in an (n+1)-dimensional Lorentzian space form ${M}_{1}^{n+1}\left(c\right)$ with non-zero constant $k$-th $(k<n)$ mean curvature and two distinct principal curvatures $\lambda $ and $\mu $. We give some characterizations of Riemannian product ${H}^{m}\left({c}_{1}\right)\times {M}^{n-m}\left({c}_{2}\right)$ and show that the Riemannian product ${H}^{m}\left({c}_{1}\right)\times {M}^{n-m}\left({c}_{2}\right)$ is the only complete connected and oriented space-like hypersurface in ${M}_{1}^{n+1}\left(c\right)$ with constant $k$-th mean curvature and two distinct principal curvatures, if the multiplicities of...