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.

We define generalized semi-invariant submanifolds in locally product Riemannian manifolds. Then we study multiply warped product generalized semi-invariant submanifolds in the same structure. We give an existence theorem for such submanifolds. We also give necessary and sufficient conditions for such a submanifold to be a multiply direct product submanifold. Moreover, we establish a general inequality for such submanifolds.

We introduce the notion of conformal-twisted product submanifolds of the formwhere M T is a holomorphic submanifold and M θ is a proper slant submanifold of M in a globally conformal Kaehler manifold and f and b are conformal factor and twisting function, respectively. We give necessary and sufficient conditions for proper semi-slant submanifold to be a locally conformal-twisted product for such submanifolds of the form f M T × b M θ and f M θ × b M T . We establish a general inequality for the squared norm of second fundamental form of these types of submanifolds.

The aim of this paper is to characterize some equations of structures for gradient almost η-Ricci Bourguignon solitons which generalize the equivalent for gradient η-Ricci-Bourguignon solitons. We prove that a gradient almost η-Ricci-Bourguignon soliton is gradient almost 1/ωu-traceless Ricci soliton with a well defined potential function f. Moreover we investigate that an Einstein manifold of constant scalar curvature is isometric to a space form with a well defined potential function f. Finally, we derived an integral formula for gradient compact case. We show that a compact nontrivial almost 1/ωu-traceless Ricci soliton with constant scalar curvature or conformal potential vector field, is isometric to a standard unit sphere, hyperbolic space and Euclidean space.

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