Global Analysis: An Introduction to Nonlinear Analysis and Its Variational Methods on Riemannian Manifolds

This monograph develops an introduction to global analysis centered on the interaction between differential geometry, functional analysis, partial differential equations, and variational methods on Riemannian manifolds. Beginning with smooth and Riemannian geometry, it develops Sobolev spaces, distributions, interpolation and fractional regularity, differential and pseudodifferential operators on vector bundles, elliptic theory, heat methods, bounded geometry, and trace theorems. It then treats Fredholm and index theory, culminating in the Atiyah–Singer index theorem, followed by geometric evolution equations and Ricci flow, infinite-dimensional geometry on Banach and Hilbert manifolds, and variational methods including the direct method, Palais–Smale theory, deformation arguments, the mountain pass theorem, and the Nehari method. Particular emphasis is placed on explicit proofs, the passage from local Euclidean estimates to intrinsic global statements, and the precise geometric hypotheses required in compact, noncompact, and boundary settings. The text is intended for advanced undergraduate and graduate students, as well as readers approaching global analysis from geometry or differential equations.