On doubly warped products

Abstract

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 sufficient 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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Keywords

• Doubly warped product
• warped product
• direct product
• Einstein manifold

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