Course Identification

Geometric structures and generalized geometry

Lecturers and Teaching Assistants

Dr. Roberto Rubio

Course Schedule and Location

First Semester
Sunday, 09:15 - 12:00, Ziskind, Rm 1

Field of Study, Course Type and Credit Points

Mathematics and Computer Science: Lecture; Elective; Regular; 3.00 points
Physical Sciences: Lecture; Elective; Regular; 4.00 points


This course makes use of the e-learning platform Perusall and active participation of the students, both online and in-class, is expected. The course contents are covered by the students before each session using Perusall, where they can read the lecture notes, share their comments, ask questions and engage in conversations. The in-class time, apart from answering questions and solving doubts, is devoted to understanding the main ideas, going beyond them (with some activities and hands-on sessions) and anticipating what comes next.

Attendance in the grade breakdown stands for active participation. The online participation will be graded by the AI system of the e-learning platform Perusall (involvement and interaction through comment posting are considered). This grade will be combined with a grade from the in-class participation.

The seminar consists of an investigation of a question related to the course and of interest to the student. The student should tackle this question critically and consulting several sources, write a very short summary of her/his findings (~2 pages) and share it with the class in a 15-min presentation.

Enter or register at (free) and use the code RUBIO-JRMKN.





Language of Instruction


Attendance and participation

Expected and Recommended

Grade Type

Pass / Fail

Grade Breakdown (in %)


Evaluation Type


Scheduled date 1


Estimated Weekly Independent Workload (in hours)



Geometric structures allow us to understand better any kind of mathematics. Once we know a space is endowed with a particular structure, we immediately gain a lot of information, a set of tools and better insights.

This course combines a scoping review on geometric structures in a broad sense (especially focusing on complex, symplectic and Kähler geometry) with an introduction to generalized geometry, a very recent approach to geometric structures that has become an active research field.

The course consists of six sections:

  1. Smooth manifolds and (G,X)-structures: definitions and examples, geometries for surfaces and three-manifolds.
  2. Fibre bundles and G-structures: the tangent bundle, the frame bundle, motivation through metrics and almost complex structures, integrability.
  3. Geometric structures through differential forms, multivector fields and distributions: presymplectic, Poisson, contact and Engel structures.
  4. Complex, symplectic and Kähler geometry: topological constraints, main theorems, and open questions.
  5. Generalized geometry: the generalized tangent bundle, the Dorfmann bracket, Dirac structures and spinors.
  6. Generalized complex and Kähler geometry: definitions and examples, type-changing structures and applications.

Learning Outcomes


Upon successful completion of this course, students will be able to:

  • Understand and connect the main ideas and motivations behind the theory.
  • Differentiate the various approaches to geometric structures.
  • Initially assess whether a given space may admit a geometric structure.
  • Deal with new geometric structures in a competent way.
  • Approach new literature on generalized geometry in a critical manner.

Reading List

Lecture notes will be provided. They will be complemented by additional references. Some of them are the following.


  • Hitchin, N. Differentiable manifolds. Lecture notes, 2014. Available at
  • Lee, J. M. Introduction to smooth manifolds. Graduate Texts in Mathematics, 218. Second Edition. Springer, New York, 2013.
  • Thurston, W.P. Three-dimensional geometry and topology. Vol. 1. Edited by Silvio Levy. Princeton Mathematical Series, 35. Princeton University Press, 1997.
  • Kobayashi, S.; Nomizu, K. Foundations of differential geometry. Vol. I. Reprint of the 1963 original. Wiley Classics Library. John Wiley & Sons, 1996.
  • McDuff, D., Salamon, D. Introduction to symplectic topology. Third edition. Oxford Graduate Texts in Mathematics. Oxford University Press, Oxford, 2017.
  • Huybrechts, D. Complex geometry. An introduction. Universitext. Springer-Verlag, Berlin, 2005.
  • Joyce, D. Slides for a graduate course on complex and Kähler geometry, 2019.
  • Rubio, R. Generalized geometry, an introduction through linear algebra. Book preprint.
  • Gualtieri, M. Generalized complex geometry. Ann. of Math.,174(1):75–123, 2011.
  • Gualtieri, M. Generalized Kähler geometry. Comm. Math. Phys. 331, no. 1, 297–331, 2014.