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Riemannian Geometry

Code: 100115
Credits: 6
2026/2027
Degree programme Type Course
Mathematics OP 4

Contact lecturer

Name :
Eduardo Gallego Gómez
Email :
eduardo.gallego@uab.cat

Teaching staff

Gil Solanes Farres

Group languages

You can consult this information at the end of the document.

Prerequisites

To successfully follow this course, students are encouraged to have acquired the fundamental knowledge of Differential Geometry. The course will also require the use of concepts and techniques from mathematical analysis (multivariable calculus and optimization), topology, and differential equations, corresponding to the courses Topology and Differential Equations and Modelling I.

Objectives

A Riemannian manifold is a differentiable manifold endowed with a positive-definite inner product on the tangent space at each point. Riemannian geometry studies these structures and originated as a generalization of the intrinsic geometry of surfaces. Over time, it has become a fundamental tool in both the formulation of classical mechanics and, especially, the theory of general relativity. More recently, it has also played a prominent role in the resolution of the Poincaré Conjecture.

The two central notions of Riemannian geometry are curvature and geodesics. The main goal of this course is to understand, from a geometric and as intuitive a perspective as possible, the relationship between these two concepts. In particular, we will study how curvature influences the behavior of geodesics and how it affects the global topological properties of manifolds.

Learning outcomes

  1. Effectively use bibliographies and electronic resources to obtain information.
  2. Students must have and understand knowledge of an area of study built on the basis of general secondary education, and while it relies on some advanced textbooks it also includes some aspects coming from the forefront of its field of study.
  3. Develop critical thinking and reasoning and know how to communicate it effectively, both in one's own languages and in a third language.
  4. Students must be capable of collecting and interpreting relevant data (usually within their area of study) in order to make statements that reflect social, scientific or ethical relevant issues.
  5. Students must be capable of communicating information, ideas, problems and solutions to both specialised and non-specialised audiences.
  6. Students must develop the necessary learning skills to undertake further training with a high degree of autonomy.
  7. Generate innovative and competitive proposals for research and professional activities.
  8. Actively demonstrate high concern for quality when defending or presenting the conclusions of one's work.
  9. Apply critical spirit and thoroughness to validate or reject both one's own arguments and those of others.
  10. Understand abstract language and in-depth demonstrations of some advanced theorems of geometry and topology.
  11. Devise demonstrations of mathematical results in the field of geometry and topology.

Contents

  1. Differentiable manifolds: manifolds, differentiable maps, tangent spaces, the tangent bundle, immersions and submanifolds, orientation, vector fields, tensors, and differential forms.
  2. Riemannian manifolds: Riemannian metrics, length of curves, distance, volume, conformal maps, and connections.
  3. Geodesics: the Levi-Civita connection, parallel transport, geodesics, the exponential map, normal coordinates, and the Gauss lemma.
  4. Curvature: the Riemann curvature tensor, associated curvature notions, and local expressions of curvature.
  5. Additional topics: examples and applications, including hyperbolic geometry and the interplay between geodesics, curvature, and topology.

Learning activities and methodology

Title Hours ECTS Learning outcomes
Preparation and exhibition of works 16 0.64 1, 5, 6, 9, 10
Theoretical course 30 1.2 6, 9, 10
Seminars 6 0.24 5, 6, 9, 10, 11
Tutorials 14 0.56 6, 9, 10, 11
Personal study 45 1.8 1, 6, 9, 10
Resolution of problems 30 1.2 1, 5, 6, 9, 11

Lectures

  • Introduction to the fundamental concepts of Riemannian geometry.
  • Presentation of the main results and theorems of the subject.
  • Development of the theoretical tools required for problem solving.
  • Study of proofs and representative examples.

Problem sessions

  • Solution of problems related to the theoretical contents.
  • Reinforcement and deeper understanding of concepts introduced in the lectures.
  • Development of computational skills and techniques specific to Riemannian geometry.
  • Discussion of different methods and problem-solving strategies.
  • Active student participation in the analysis and solution of exercises.

Seminars

  • In-depth study of selected topics from the course.
  • Independent work based on material provided in advance.
  • Solution of theoretical questions and more challenging problems.
  • Guidance and support from the instructors during the sessions.
  • Submission and assessment of the assigned work.

Individual project

  • Preparation of a project on a topic related to the course.
  • Writing of a short report.
  • Oral presentation of the project in class.
  • Development of independent learning, synthesis, and mathematical communication skills.

Tutorials

  • Resolution of questions concerning both theoretical and practical aspects of the course.
  • Individualized guidance on coursework and projects.
  • Monitoring of students’ academic progress.
Annotation: within the schedule set by the centre or degree programme, 15 minutes of one class will be reserved for students to evaluate their lecturers and their courses or modules through questionnaires.

Assessment

Continuous assessment activities

Title Weight Hours ECTS Learning outcomes
Delivery of problems 0,3 4 0.16 1, 2, 5, 6, 8, 9, 10, 11
Exam 0,4 4 0.16 1, 5, 6, 8, 9, 10, 11
Presentation of works 0,3 1 0.04 1, 2, 3, 4, 5, 6, 7, 9, 10

Continuous Assessment

Assessment in this course will take into account both the assimilation of the course content and the work carried out throughout the semester. Evaluation will be based on a system of continuous assessment.

The final grade will be calculated as a weighted average of the following assessment components:

  • Examination component: 40%.
  • Problem sets and assignments component: 30%.
  • Project and oral presentation component: 30%.

Any honours distinctions (Matrícula de Honor) will be awarded on the basis of the final continuous assessment grade.

Students who do not pass the continuous assessment (i.e., who obtain a final grade below 5.0 out of 10) or who wish to improve their grade may take a resit examination.

A student will be recorded as “Not Assessed” if the weight of the assessment activities in which they have participated does not exceed 50% of the total assessment for the course.

Single Assessment

Students who have opted for the single-assessment modality must complete a final examination covering the contents of the course.

This examination will take place on the same date, at the same time, and in the same location as the examination scheduled for the continuous-assessment modality. Upon completion of the examination, students must submit the projects and assignments required for continuous assessment.

The final grade will be calculated as a weighted average of the following components:

  • Final examination: 40%.
  • Projects and assignments: 60%.

If the final grade is below 5.0 out of 10, students will have a further opportunity to pass the course by taking the resit examination, which will be held on the date established by the degree programme coordinator. The projects and assignments component is not recoverable and will therefore retain the original grade obtained.

Bibliography

  1. Manfredo P. do Carmo, Riemannian Geometry. Birkhäuser, 1992.
  2. Manfredo P. do Carmo, Geometría diferencial de curvas y superficies. Alianza Universidad, 1990.
  3. S. Gallot, D. Hulin i J. Lafontaine, Riemannian Geometry. Springer-Verlag, 1990.
  4. Joan Girbau, Geometria diferencial i relativitat. Manuals de la UAB, Servei de Publicacions de la Universitat Autònoma de Barcelona, 1993.
  5. John M. Lee, Riemannian Manifolds: An Introduction to Curvature. Springer-Verlag, 1997.
  6. M. Spivak, A Comprehensive Introduction to Differential Geometry. Publish or Perish, 1979.
  7. J. Cheeger i D. Ebin, Comparison Theorems in Riemannian Geometry. North-Holland, 1975.

Software

The use or learning of any specific software is not required to successfully complete the course. All course content and assessment activities can be followed and completed using the mathematical tools developed throughout the course.

Nevertheless, the use of SageManifolds (<https://sagemanifolds.obspm.fr>) within the SageMath mathematical computing environment (<https://www.sagemath.org>) is recommended, as it provides a useful tool for experimentation, visualization, and computations in differential and Riemannian geometry. Its use is entirely optional and intended solely as a supplementary resource.

Course groups and languages

The information provided is provisional until November 30. After this date, you will be able to consult the language of each group through this link. To access the information, you will need to enter the course CODE

Type of teaching Group Language Semester Shift
(TE) Theory 1 Catalan second semester morning-mixed
(PAUL) Classroom practices 1 Catalan second semester morning-mixed
(SEM) Seminars 1 Catalan second semester afternoon