Supersymmetric dS _n solutions for n ⩾ 5in D = 11 supergravity

We determine the necessary and sufficient conditions for warped product $dS_n$ solutions, $5 \leq n 10$, to preserve supersymmetry in $D=11$ supergravity, without assuming factorization of Killing spinors. prove that $7 all such solutions are flat, with vanishing 4-form. also show only $dS_6$ either maximally supersymmetric $AdS_7 \times S^4$ solution, or $\mathbb{R}^{1,6} N_4$ where $N_4$ is hyperKahler, Supersymmetric $dS_5$ then classified; it shown generalized M5-brane configurations, which transverse space $\mathbb{R} N_4$, a hyperKahler manifold. If 4-form covariantly constant, admits potential.