A section in ℝ³U × ℝ → U
01 / VECTOR BUNDLES
From a fiber to a section
A family of vector spaces over a manifold; a section chooses a vector in each fiber.
MATHEMATICS · UNAM & ANÁHUAC MÉXICO
Carlos Daniel
Velázquez Mendoza
Analysis through the lens of geometry.
Ph.D. student in Mathematics at UNAM. I study Sobolev spaces, geometric PDE, and variational methods on manifolds and vector bundles.
01 / LOCAL → GLOBAL GEOMETRY𝕋² ⊂ ℝ³
Global analysisDifferential geometrySobolev spacesNonlinear analysisGeometric PDEVariational methods
M.Sc. in Mathematics
UNAM · Mexico City
ACADEMIC PROFILE
Global analysis and geometry
I am a Ph.D. student in the Graduate Program in Mathematical Sciences at Universidad Nacional Autónoma de México (UNAM). I study Sobolev spaces and partial differential equations on Riemannian manifolds and vector bundles.
My work uses differential geometry, functional analysis and variational methods to study these problems through intrinsic formulations. I am also interested in general topology, particularly realcompact spaces.
I teach at the Faculty of Sciences of UNAM and at Universidad Anáhuac.
A GEOMETRIC ATLAS
Geometry and analysis.
A trace at the boundaryTr u = u|∂Ω
02 / TRACES & BOUNDARY
Analysis reaches the boundary
For smooth functions, the trace recovers boundary values. Sobolev theory extends this idea.
A search for critical pointsγ̇ = −∇J
03 / VARIATIONAL METHODS
The geometry of an energy
An energy landscape with two minima illustrates the search for critical points.
Academic profile
Research
My research lies in global analysis, geometric analysis, Sobolev spaces on manifolds and vector bundles, geometric PDE, differential operators, and variational methods.
Undergraduate teaching
I have taught undergraduate courses at the Faculty of Sciences of UNAM and at Universidad Anáhuac. I am currently teaching at both institutions.
Graduate-level support
I have assisted in graduate courses on nonlinear analysis on Riemannian manifolds and variational methods on Riemannian manifolds.
Mathematical exposition
I develop notes, guides, interactive tools, and manuscripts in calculus, global analysis, differential equations, differential geometry, and functional analysis.
Recent publications
NODEA · 2026
Integration by parts formula and Sobolev embeddings on vector bundles: an intrinsic geometrical global approach
With María de los Ángeles Sandoval-Romero.
NoDEA 33, 133 (2026).
ARXIV · 2026
An Intrinsic Sobolev Trace Theory for Riemannian Vector Bundles with Applications to Quasilinear Elliptic Equations
With María de los Ángeles Sandoval-Romero and Romulo Diaz Carlos.
Books
Global Analysis
An introduction to nonlinear analysis and its variational methods on Riemannian manifolds.
With María de los Ángeles Sandoval Romero and Romulo Diaz Carlos. Spanish and English editions.
NEW BOOK
Cálculo Diferencial e Integral en Variedades
Del cálculo multivariable al cálculo en variedades.
Multivariable calculus, integration and geometry, with applications and interactive material. First edition, 2026. Available in Spanish.
Current research
I am currently working on maximum principles for quasilinear equations on vector bundles and Kirchhoff–Boussinesq problems with Navier boundary conditions.