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
From local structure to global questions
I am a Ph.D. student in the Graduate Program in Mathematical Sciences at Universidad Nacional Autónoma de México (UNAM). My academic interests include teaching and research in global analysis, partial differential equations on Riemannian manifolds and vector bundles, variational calculus on manifolds and vector bundles, geometric analysis, and topology, particularly realcompact spaces.
My current work focuses on developing intrinsic results in global analysis, connecting differential geometry, functional analysis, Sobolev theory, PDE, and variational methods. I am especially interested in geometric formulations of PDE problems on vector bundles.
I teach at the Faculty of Sciences of UNAM and at Universidad Anáhuac.
A GEOMETRIC ATLAS
Three perspectives, one language.
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.
Featured work
Geometric Integration by Parts and Sobolev Spaces on Vector Bundles
With María de los Ángeles Sandoval Romero.
arXiv:2602.01016, 2026.
Global Analysis
An introduction to nonlinear analysis and its variational methods on Riemannian manifolds.
First preliminary version · 2026, with María de los Ángeles Sandoval Romero and Romulo Diaz Carlos.
Research profile · ORCID preview
ORCID · 0009-0004-1063-8195
Selected 2026 output
A quick preview of the publications highlighted on this site. The ORCID profile remains the canonical external research identifier.
2026 · PREPRINT
Geometric Integration by Parts and Sobolev Spaces on Vector Bundles: A Unified Global Approach
arXiv:2602.01016
Open on arXiv ↗
2026 · BIRKHÄUSER
A Simple Proof of Metric Independence of Sobolev Norms on Compact Manifolds
Analysis and PDE in Latin America, Trends in Mathematics, vol. 15.
Open on Springer ↗
Current project
Ground state solutions for quasilinear equations on vector bundles
Work in progress with María de los Ángeles Sandoval Romero and Romulo Diaz Carlos on quasilinear elliptic equations on vector bundles from the perspective of geometric analysis and variational methods. The project studies intrinsic formulations through the formal adjoint of the covariant derivative and divergence-form problems on Riemannian manifolds with boundary.