Books & Notes
First preliminary version · 2026
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GLOBAL ANALYSIS
32 chapters · 7 parts · 4 appendices
FIRST PRELIMINARY VERSION · 2026
Global Analysis
An introduction to nonlinear analysis and its variational methods on Riemannian manifolds
Carlos Daniel Velázquez Mendoza, María de los Ángeles Sandoval Romero, and Romulo Diaz Carlos.
A self-contained path from differential geometry and functional analysis to Sobolev theory on manifolds and vector bundles, differential and pseudodifferential operators, index theory, geometric evolution equations, infinite-dimensional geometry, and variational methods.
The book is designed so that the reader can follow the passage from Euclidean estimates to intrinsic geometric formulations without hiding localization, boundary, or bundle arguments behind black boxes.
First preliminary version. This page hosts the current English manuscript. The book remains under development: exercises and other complementary material are still to be added. All rights reserved.
What the book develops
From local to global
Charts remain a computational tool, but the emphasis is on identifying which constructions are intrinsic and on making the local-to-global passage explicit.
Sobolev theory on bundles
Weak covariant derivatives, formal adjoints, integration by parts, density, embeddings, compactness, traces, fractional scales, and bounded geometry.
Geometry meets PDE
Elliptic operators, pseudodifferential calculus, index theory, Ricci flow, and variational formulations of nonlinear equations on Riemannian manifolds.
General structure
I. Riemannian manifolds
Smooth manifolds, vector bundles, Riemannian metrics, connections, parallel transport, geodesics, curvature, integration, and Clifford geometry.
II. Nonlinear analysis in Euclidean spaces
Weak derivatives, Sobolev spaces, Rellich–Kondrashov, distributions, Fourier analysis, interpolation, Bessel potentials, Slobodeckij, Besov, Triebel–Lizorkin, and trace theory.
III. Nonlinear analysis on Riemannian manifolds
Sobolev spaces on manifolds and vector bundles, formal adjoints, distributions, elliptic operators, heat kernels, fractional spaces, bounded geometry, and traces.
IV. Index theory
Pseudodifferential operators, Fredholm theory, K-theory, Bott periodicity, the Thom isomorphism, and the Atiyah–Singer index theorem.
V. Geometric evolution equations
Geometric parabolic equations, Ricci flow, compactness, singularities, and Perelman’s theory.
VI. Infinite-dimensional geometry
Banach manifolds, bundles, connections, Hilbert and Finsler geometry, Sobolev mapping spaces, gauge theory, and diffeomorphism groups.
VII. Variational methods on Riemannian manifolds
Weak formulations, the direct method, Palais–Smale, pseudogradients, deformation, mountain-pass methods, Nehari constraints, and quasilinear equations on vector bundles.
Appendices
Topology, mathematical analysis, linear algebra, and the Einstein summation convention.
Read the book online
Notes
Calculus notes
Coming soon. This section will include notes on differential and integral calculus, multivariable calculus, optimization, and geometric aspects of calculus.
Companion material
Shorter notes, figures, exercises, and interactive material related to the book can be added here without changing the main edition.