Books & Notes
Book project
Análisis global
Una introducción al análisis no lineal y sus métodos variacionales en variedades riemannianas.
Carlos Daniel Velázquez Mendoza, María de los Ángeles Sandoval Romero, and Romulo Diaz Carlos.
This manuscript is currently a work in progress. Its goal is to offer a modern, rigorous, and geometrically motivated introduction to global analysis and nonlinear analysis on Riemannian manifolds and vector bundles.
The text begins with the foundations of differential geometry and functional analysis, and then moves toward Sobolev spaces, distributions, differential operators, index theory, geometric PDE, and variational methods. The aim is to build a self-contained and progressive exposition for advanced undergraduates, graduate students, and researchers interested in the interaction between analysis, geometry, and PDE.
Copyright notice. This manuscript is a work in progress. All rights reserved by the authors. A version of the book is planned for publication at UNAM.
Motivation
Nonlinear analysis on Riemannian manifolds lies at one of the richest intersections of contemporary mathematics: differential geometry, functional analysis, and partial differential equations. The manuscript aims to show how classical analytic tools are transformed when they are transferred to curved spaces, where the metric, curvature, topology, and global structure directly influence analytic properties.
A guiding theme of the text is the passage from calculus and linear algebra to nonlinear analysis: from local linear approximation through derivatives and tangent spaces to weak and variational formulations in infinite-dimensional spaces.
General structure
I. Riemannian manifolds
Smooth manifolds, vector bundles, Riemannian metrics, connections, geodesics, normal and Fermi coordinates, curvature, integration, and common differential operators on manifolds.
II. Nonlinear analysis in Euclidean spaces
Weak derivatives, Sobolev spaces, embeddings, Rellich–Kondrashov, distributions, LF spaces, Schwarz class, Fourier transform, fractional spaces, Besov and Triebel–Lizorkin spaces, traces, and interpolation.
III. Nonlinear analysis on Riemannian manifolds
Sobolev spaces on manifolds, Kato inequality, embeddings, compactness, operators on vector bundles, formal adjoints, Sobolev spaces on vector bundles, distributions, heat kernels, and bounded geometry.
IV. Index theory
Pseudodifferential operators, elliptic operators, Fredholm theory, topological K-theory, and the Atiyah–Singer index theorem.
V. Applications
Partial differential equations on Riemannian manifolds, Ricci flow, Sobolev spaces, and applications to geometric analysis.
VI. Infinite-dimensional analysis
Calculus in infinite-dimensional spaces and infinite-dimensional manifolds.
VII. Variational methods
Weak and strong solutions, differentiability of functionals, variational methods, nonlinear growth, minimization, and Lagrange multipliers in Banach spaces.
Appendices
Topology, analysis, metric spaces, normed spaces, Hilbert spaces, measure theory, \(L^p\) spaces, topological vector spaces, weak topologies, linear algebra, and tensor notation.
Manuscript preview
Notes
Calculus notes
Coming soon. This section will include notes on differential and integral calculus, multivariable calculus, optimization, and geometric aspects of calculus.
Global analysis notes
The book manuscript above is the main ongoing project in global analysis. Additional shorter notes may be added later.