Directed Acyclic Graphs and trees are widely prevalent in several real-world applications. These hierarchical structures show intriguing properties such as scale-free and bipartite nature, with fine-grained temporal irregularities among nodes. Building on advances in geometrical deep learning, we explore a time-aware neural network to model trees and Directed Acyclic Graphs in multiple Riemannian manifolds of varying curvatures. To jointly utilize the strength of these manifolds, we propose Multi-Manifold Recursive Interaction Learning (MRIL) on Directed Acyclic Graphs where we introduce an inter-manifold learning mechanism that recursively enriches each manifold with representations from sibling manifolds. We propose the integration of the Stiefel orthogonality constraint which stabilizes the training process in Riemannian manifolds. Through a series of quantitative and exploratory experiments, we show that our method achieves competitive performance and converges much faster on data spanning several domains.