Geometric Analysis of Riemannian Doubly Warped Product Submersions

This paper introduces the concept of Riemannian doubly warped product submersions as a natural generalization of Riemannian warped product submersions. We study some fundamental properties of such submersions and derived the conditions under which the Riemannian doubly warped product submersions have totally geodesic, totally umbilical, and minimal fibers. Furthermore, we analyze the curvature properties associated with Riemannian doubly warped product submersions and establish the Einstein conditions. Some non-trivial examples are explored to demonstrate the existence of such submersions.