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2020/2021

Riemannian geometry

Code: 100115 ECTS Credits: 6
Degree Type Year Semester
2500149 Mathematics OT 4 0
The proposed teaching and assessment methodology that appear in the guide may be subject to changes as a result of the restrictions to face-to-face class attendance imposed by the health authorities.

Contact

Name:
Florent Balacheff
Email:
Florent.Balacheff@uab.cat

Use of Languages

Principal working language:
catalan (cat)
Some groups entirely in English:
No
Some groups entirely in Catalan:
Yes
Some groups entirely in Spanish:
No

Prerequisites

For good follow-up of the subject it is recommended a good assimilation of the concepts introduced in the subject Geometría Diferencial.
Analysis knowledge (Calculation in various variables and optimization), topology (Topology) and differential equations (Differential Equations and Modeling I) will also be used.

Objectives and Contextualisation

A Riemannian manifold is a differentiable manifold endowed with a scalar product in the tangent space at each point. Riemannian geometry is the study of Riemannian manifolds and first appears as a generalization of the intrinsic geometry of surfaces. Later it turns out to be the perfect paradigm where classical mecanic and general relativity can be formulated. More recently this topic was decisive in the proof of Poincaré's conjecture albeit the topological nature of this conjecture.


The two main notions in Riemannian geometry are curvature and geodesics. The main objective of this course is to understand, geometrically and as far as possible, the interconnexion between these two notions. In particular we will analyze the effect of curvature on geodesics and on the topology of manifolds.

Competences

  • Actively demonstrate high concern for quality when defending or presenting the conclusions of one’s work.
  • Apply critical spirit and thoroughness to validate or reject both one’s own arguments and those of others.
  • Demonstrate a high capacity for abstraction.
  • Develop critical thinking and reasoning and know how to communicate it effectively, both in one’s own languages and in a third language.
  • Effectively use bibliographies and electronic resources to obtain information.
  • Formulate hypotheses and devise strategies to confirm or reject them.
  • Generate innovative and competitive proposals for research and professional activities.
  • 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.
  • Students must be capable of communicating information, ideas, problems and solutions to both specialised and non-specialised audiences.
  • Students must develop the necessary learning skills to undertake further training with a high degree of autonomy.
  • 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.

Learning Outcomes

  1. Actively demonstrate high concern for quality when defending or presenting the conclusions of one’s work.
  2. Apply critical spirit and thoroughness to validate or reject both one’s own arguments and those of others.
  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. Devise demonstrations of mathematical results in the field of geometry and topology.
  5. Effectively use bibliographies and electronic resources to obtain information.
  6. Generate innovative and competitive proposals for research and professional activities.
  7. 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.
  8. Students must be capable of communicating information, ideas, problems and solutions to both specialised and non-specialised audiences.
  9. Students must develop the necessary learning skills to undertake further training with a high degree of autonomy.
  10. 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.
  11. Understand abstract language and in-depth demonstrations of some advanced theorems of geometry and topology.

Content

1. Riemannian manifolds. Notion of Riemannian length and volume.
2. Connections. Geodesics. Exponential map and Gauss Lemma. The Hopf-Rinow theorem.
3. Curvature. Jacobi fields.
4. Hyperbolic geometry.
5. Introduction to systolic geometry.

Methodology

The subject has two weekly hours of theory class and one of problems. In addition, throughout the course there will be three seminars of two hours each.
In the theory classes, the fundamental notions of Riemannian geometry will be introduced and the most important results of the theory will be presented. Likewise, the necessary tools for the understanding and resolution of problems will be given.
The problems classes will be deepened in the assimilation and the understanding of the concepts developed in the theoretical classes will be improved by solving theoretical problems and exercises designed to increase the students' dexterity in their own calculations. This work will be carried out through explanations made by the teacher on the board and the active participation of students in the discussion of the different arguments used to solve the problems. The lists of problems will be delivered to the students throughout the semester.
The seminars will be devoted to deepening issues dealt with in theory classes and problems. Students will receive a script before each seminar will be held. During the session, they will have to work autonomously, although they can be consulted by the teachers. Subsequently, they will deliver the solution to the problems worked during the seminar. These solutions will be corrected by the teachers, giving rise to a part of the continuous evaluation note.
At the same time, each student will draw up a work on a chosen subject from a list proposed by the teachers. This work will be delivered in writing, as well as exposing yourself in class. The evaluation of both aspects (written and oral) will also be part of the continuous evaluation.
Individual tutorials, or in small groups, are foreseen for students who want it in the professor's office.
In the end the student will have received the theory and problems classes, as well as the seminars, all the necessary information (both the statements and their demonstrations), to face the partial test such as the final test. This subject will also offer resources through the Virtual Campus.

Activities

Title Hours ECTS Learning Outcomes
Type: Directed      
Theoretical course 30 1.2 2, 11, 9
Tutorials 14 0.56 2, 11, 4, 9
Type: Supervised      
Seminars 6 0.24 2, 11, 4, 9, 8
Type: Autonomous      
Personal study 45 1.8 2, 11, 9, 5
Preparation and exhibition of works 16 0.64 2, 11, 9, 8, 5
Resolution of problems 30 1.2 2, 4, 9, 8, 5

Assessment

The evaluation of this subject will take into account the assimilation of the contents, as well as the work done during the course, and will be carried out in the form of a continuous evaluation. The final grade will be obtained by weighted average between the score obtained in the partial exam module (30%), the final exam module (30%), the problem delivery module (20%) and the presentation module of work (20%). The possible license plates will be awarded based on the continuous evaluation note. Students who had not passed the continuous evaluation, that is to say that they had not obtained a final grade equal or superior to five, or that they want to improve their mark, will have a final test of recovery of the modules of examinations and of delivery of problems.
A student will be qualified as "Not Presented" if the weight of the assessment activities in which he has participated does not exceed 50% of the weight of the continuous assessment of the subject.

Assessment Activities

Title Weighting Hours ECTS Learning Outcomes
Delivery of problems 0,20 1 0.04 2, 11, 1, 4, 10, 9, 8, 5
Final exam 0,30 1.5 0.06 2, 11, 1, 4, 9, 8, 5
Partial exam 0,30 1.5 0.06 2, 11, 1, 4, 9, 8, 5
Presentation of works 0,20 1 0.04 2, 11, 3, 6, 10, 9, 8, 7, 5
Recovery test 0,80 4 0.16 2, 11, 1, 4, 9, 8, 5

Bibliography

1- Manfredo P. do Carmo, Riemannian Geometry. Birkhäuser, 1992.

2- Manfredo P. do Carmo, Geometría diferencial de curvas y superfícies. Alianza Universidad, 1990.

3- S. Gallot, D. Hulin, J. Lafontaine, Riemannian Geometry. Springer-Verlag, 1990.
4- Joan Girbau, Geometria diferencial i relativitat. Manuals de la UAB, Servei de Publicacions de la U.A.B.,1993.
5- John M. Lee, Riemannian Manifolds: An introduction to curvature. Springer-Verlag, 1997.

6- M. Spivak, A Comprehensive Introduction to Diferential Geometry. Publish or Perish Inc, 1979.