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This paper presents a series of constructions providing eleven-dimensionalbosonic supergravity backgrounds. In particular, we treat Lorentzian manifoldsgiven in terms of twisted products of six-dimensional Lorentzian manifolds andfive-dimensional Riemannian manifolds. By considering a representative flux4-form adapted to this setting, we analyse the system of bosonic supergravityequations and describe the corresponding geometric constraints. The newsupergravity backgrounds appear for special cases associated to the adaptedflux 4-form. For example, we provide a relation of eleven-dimensionalsupergravity with Ricci-isotropic Walker manifolds, and illustrate severalresults in their terms and in terms of Ricci-flat Riemannian manifolds.

Decomposable (6, 5)-solutions in eleven-dimensional supergravity