We give a new characterization for doubly warped products by using the geometry of their canonical foliations intersecting perpendicularly. We also give a necessary and su cient condition for a doubly warped product to be a warped or a direct product. As a result, we prove the non-existence of Einstein proper doubly warped product pseudo-Riemannian manifold of dimension grater or equal than 4.
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Aydın et al. (2020) studied this question.
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