Let \(E \rightarrow B\) be a fibration such that \(E\) and \(B\) are Kähler manifolds, the projection is a conformal submersion, and all fibers are totally geodesic pairwise isomorphic submanifolds. The author proposes to consider such an object as an analog of a warped product for Kähler manifolds. He proves that, for every complete Hodge manifold, there exists such a fibration which is a complete manifold itself. Some other results on this subject are also obtained.