Decomposability of Quadratic Killing Tensors on Rank-One Symmetric Spaces
2026A. K. D. Nguyen
Journal of Differential Geometry (to appear)
Department of Mathematics
An Ky Duy Nguyen— Kyan
PhD Student in Differential Geometry & Riemannian Geometry
I am a PhD student working in differential geometry and Riemannian geometry. My current research focuses on the decomposability of quadratic Killing tensors on symmetric spaces. More broadly, I am interested in geometric structures, curvature, symmetric spaces, integrable systems, and the interaction between geometry, topology, and mathematical physics.
About
A short introduction
- Affiliation
- Department of Mathematics
- kyan@kyanduynguyen.com
I am a PhD student working in differential geometry and Riemannian geometry. My current research focuses on the decomposability of quadratic Killing tensors on symmetric spaces. More broadly, I am interested in geometric structures, curvature, symmetric spaces, integrable systems, and the interaction between geometry, topology, and mathematical physics.
Trained in pure mathematics, I approach geometry through the interplay of curvature and symmetry. I am drawn to problems where classical Riemannian geometry meets the algebraic structure of Lie groups and the analytic subtlety of integrable systems. Outside research, I keep a growing set of typeset notes on the topics I return to most often.
A recurring object in my work is a Killing tensor $K$ of order two, satisfying $\nabla_{(a} K_{bc)} = 0$, and its decomposition into symmetric products of Killing vector fields on a symmetric space $G/H$.Research Interests
- Differential Geometry
- Riemannian Geometry
- Symmetric Spaces
- Killing Tensors
- Integrable Systems
- Geometric Analysis
Research
Geometry, symmetry, and hidden structure
I work at the intersection of Riemannian geometry, Lie theory, and integrable systems.
Current Project
Decomposability of Quadratic Killing Tensors on Symmetric Spaces
My current PhD research investigates the decomposability of quadratic Killing tensors on symmetric spaces. These tensors encode hidden symmetries of Riemannian manifolds and play an important role in the study of integrable geodesic flows, separation of variables, and geometric structures.
\[ K = \sum_{i} c_i\, X_i \odot Y_i, \qquad X_i, Y_i \in \mathfrak{iso}(M,g). \]
Publications
Preprints
Work in Progress
Higher-Order Killing Tensors on Non-Compact Symmetric Spaces
2026A. K. D. Nguyen
Manuscript in preparation
Talks & Activities
Seminars, conferences, and visits
A record of recent talks, workshops, and academic visits.
- March 2026Seminar Talk
Decomposability of Killing Tensors on Symmetric Spaces
Geometry Seminar, University of Oxford
An expository talk on hidden symmetries of Riemannian manifolds and the algebraic obstructions to decomposing quadratic Killing tensors.
- January 2026Winter School
Integrable Geodesic Flows and Symmetric Spaces
Winter School on Geometry & Physics, Srní
Three lectures on the interaction between symmetric-space geometry, Killing tensors, and integrability of geodesic flows.
- November 2025Conference Talk
Symmetry Reduction of Second-Order Killing Tensors
Differential Geometry Workshop, ETH Zürich
Recent progress on classifying reducible Killing 2-tensors on compact rank-one symmetric spaces.
- September 2025Academic Visit
Research visit — Institut des Hautes Études Scientifiques
IHÉS, Bures-sur-Yvette
One-month research stay working on holonomy and hidden symmetries with the geometry group.
- June 2025Seminar Talk
Killing Tensors, an Introduction
Graduate Seminar, Department of Mathematics
A gentle introduction to Killing tensors for graduate students, with worked examples on spheres and complex projective spaces.
Teaching
Teaching assistantships & tutorials
Selected teaching from recent terms, with problem sheets and lecture notes.
Michaelmas 2025Teaching Assistant
Differential Geometry (Part B)
Department of Mathematics
Trinity 2025Tutor
Analysis on Manifolds
Department of Mathematics
Hilary 2025Teaching Assistant
Introduction to Riemannian Geometry
Department of Mathematics
Notes
A working notebook
Typeset notes I return to often. Living documents — updated as I learn.
§142 pp.
Notes on Riemannian Geometry
A self-contained set of notes covering connections, curvature, geodesics, and comparison theorems.
Download PDF§58 pp.
Killing Tensors
An introduction to Killing tensors on Riemannian manifolds with worked examples on rank-one symmetric spaces.
Download PDF§96 pp.
Symmetric Spaces
Cartan's classification, root systems, and the geometry of Riemannian symmetric spaces.
Download PDF§74 pp.
Comparison Geometry
Rauch, Toponogov, and Bishop–Gromov, with an eye toward applications in Ricci-curvature bounds.
Download PDF§210 pp.
Reading Notes
Personal marginalia on Besse, Petersen, Helgason, and other standard references.
Download PDF§88 pp.
Seminar Notes
Live-typed notes from graduate seminars on integrable systems and geometric analysis.
Download PDFLatest Updates
Recent activity
- Feb 2026New preprint submitted to arXiv on reducibility of Killing 2-tensors.
- Jan 2026Invited seminar talk at Oxford Geometry Seminar.
- Dec 2025Participated in the Winter School on Geometry & Physics, Srní.
- Nov 2025Uploaded new lecture notes on comparison geometry.
Research Journey
A short chronology
- 2025 – PresentPhD in Differential Geometry
Decomposability of quadratic Killing tensors on symmetric spaces.
- 2023 – 2025MSc in Mathematics
Thesis on curvature and holonomy of Riemannian manifolds.
- 2019 – 2023BSc in Mathematics
First-class honours; foundations in analysis, topology, and geometry.
Contact
Get in touch
For research correspondence, collaboration, or teaching enquiries.
- Office
- Office 000, Mathematical Institute
- Department
- Department of Mathematics
- GitHub @kyanduynguyen
- Google Scholar Profile
- ORCID 0000-0000-0000-0000
- Curriculum Vitae Download PDF