Mathematical Foundations of Artificial Intelligence: Basics of Manifold Theory

Mathematical Foundations of Artificial Intelligence: Basics of Manifold Theory is the first volume in a two-part series.Together, they establish a unifying mathematical framework based on smooth manifold theory and Riemannian geometry·essential tools for representing, analyzing, and integrating the growing complexity of modern artificial intelligence (AI) systems and scientific models. Differential geometry now plays a central role across AI, biology, physics, and medicine.From deep learning, generative modeling, and manifold learning to reasoning algorithms and physical AI, manifolds offer a coherent geometric language that bridges theory and practice.This volume introduces key concepts·topological and smooth manifolds, Riemannian metrics, differential forms, Lie derivatives, and statistical geometry·alongside illustrative applications to data science, genomics, drug discovery, and AI-driven systems. Unlike traditional texts, this book combines rigor with intuition, integrating formal