Global Analysis: An Introduction to Nonlinear Analysis and Its Variational Methods on Riemannian Manifolds
Abstract
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.
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BibTeX
@misc{velazquezmendoza2026global,
title = {{Global Analysis: An Introduction to Nonlinear Analysis and Its Variational Methods on Riemannian Manifolds}},
author = {Carlos Daniel Velázquez-Mendoza and María de los Ángeles Sandoval-Romero and Romulo Diaz Carlos},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.14580},
eprint = {2609.14580},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14580},
}
RIS
TY - UNPB
TI - Global Analysis: An Introduction to Nonlinear Analysis and Its Variational Methods on Riemannian Manifolds
AU - Carlos Daniel Velázquez-Mendoza
AU - María de los Ángeles Sandoval-Romero
AU - Romulo Diaz Carlos
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2609.14580
UR - https://arxiv.org/abs/2609.14580
AB - 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.
ER -