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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...