Quadrotor stability under complex dynamic disturbances and model uncertainties poses significant challenges. One of them remains the underfitting problem in high-dimensional features, which limits the identification capability of current learning-based methods. To address this, we introduce a new perspective: Dimension-Decomposed Learning (DiD-L), from which we develop the Sliced Adaptive-Neuro Mapping (SANM) approach for geometric control. Specifically, the high-dimensional mapping for identification is axially ``sliced" into multiple low-dimensional submappings (``slices"). In this way, the complex high-dimensional problem is decomposed into a set of simple low-dimensional tasks addressed by shallow neural networks and adaptive laws. These neural networks and adaptive laws are updated online via Lyapunov-based adaptation without any pre-training or persistent excitation (PE) condition. To enhance the interpretability of the proposed approach, we prove that the full-state closed-loop system exhibits arbitrarily close to exponential stability despite multi-dimensional time-varying disturbances and model uncertainties. This result is novel as it demonstrates exponential convergence without requiring pre-training for unknown disturbances and specific knowledge of the model.