ECTS credits ECTS credits: 3
ECTS Hours Rules/Memories Student's work ECTS: 51 Hours of tutorials: 3 Expository Class: 9 Interactive Classroom: 12 Total: 75
Use languages Spanish, Galician
Type: Ordinary subject Master’s Degree RD 1393/2007 - 822/2021
Departments: Mathematics
Areas: Geometry and Topology
Center Faculty of Mathematics
Call: First Semester
Teaching: With teaching
Enrolment: Enrollable | 1st year (Yes)
- Introduce the student to the foundations of riemannian geometry which is a natural generalization of the theory of surfaces in Euclidean spaces. The differences between the local and global aspects of the theory will be emphasized, paying special attention to the relation between topological and analytical aspects.
- Initiate the student into the study of Lorentzian geometry, which has physical interest in the mathematical formulation of the theory of General Relativity. The differences between Riemannian and Lorentzian geometry will be particularly relevant.
- Make the student get more focused on the methods than on the concrete contents and allow him to acquire a degree of scientific maturity that enables him to tackle difficult problems, thus encouraging his hability to apply general theories to concrete situations, summarize partial results and deduce more global ones.
1 Local Riemannian geometry.
1.1. Riemannian metrics: distance functions.
1.2. Levi-Civita connection.
1.3. Geodesics and distance.
1.4. Curvature: sectional, Ricci and scalar curvatures.
1.5. Jacobi Fields: conjugate points.
1.6. Determining the metric from the curvature: Cartan’s theorem.
2 Global Riemannian geometry .
2.1. Completeness: Hopf-Rinow theorem.
2.2. Global version of Cartan’s theorem.
2.3. Complete manifolds of positive curvature: Myers’ theorem.
2.4. Complete manifolds of negative curvature: Hadamard’s theorem.
2.5. Cmparison results and applications.
3 Lorentzian and semi-Riemannian geometry
3.1. Semi-riemannian metrics: existence.
3.2. Local properties: curvature and degenerate planes.
3.3. Geodesic completeness of Lorentzian metrics.
3.4. Physical applications: relativistic space-times.
Basic bibliography
- J. M. LEE, Riemannian geometry, an introduction to curvature, Graduate Texts in Mathematics, 176. Springer-Verlag, New York, 1997.
- M. P. DO CARMO, Geometria Riemanniana, Projeto Euclides, IMPA, Rio de Janeiro, 1979.
Complementary bibliography
- J. K. BEEM, P. E. EHRLICH, K. L. EASLEY, Global Lorentzian geometry, Monographs and Textbooks in Pur. Appl. Math. 202, Marcel Dekker, Inc., New York, 1996.
- W. M. BOOTHBY, An introduction to differentiable manifolds and Riemannian geometry. Pure Appl. Math., 120. Academic Press, Florida, 1986.
- I. CHAVEL, Riemannian geometry, a modern introduction, Cambridge Tracts in Mathematics, 108. Cambridge University Press, Cambridge, 1993.
- B. O'NEILL, Semi-Riemannian Geometry with applications to relativity, Pure Appl. Math., 103. Academic Press, New York-London, 1983.
- R. K. SACHS, H. WU, General Relativity for Mathematicians, Graduate Texts in Math. 48, Springer-Verlag, New York, 1977.
- T. SAKAI, Riemannian geometry, Transactions of Mathematical Monographs 149, American Mathematical Society, Providence, RI, 1996.
Upon successful completion of this subject the student should be able to:
- Calculate the geometric objects of a Riemann manifold, such as the metric, the Levi-Civita connection or the curvature tensor.
- Determine the properties of geodesics, such as the possibility of minimizing distance and its relation with the completeness of the manifold.
- Apply the global theorems of Riemann geometry to deduce geometric and topological properties of a manifold.
- Apply Riemannian geometry and its generalizations to the theory of General Relativity.
The subject will be developed alternately through theoretical classes and practical classes encouraging the participation of the student. There will be weekly presentations, so that the student can delve into both the theoretical and practical development of the topics. Therefore, in addition to the presentations by the teacher of the different topics of the program, the student will have to develop some of the lessons throughout the course.
In addition, worksheets will be given to students on a regular basis. Some will be proposed for presentation at the conclusion of the course; the rest will be solved on the board under the supervision of the teacher. Students will also be encouraged to attend the various seminars that can be held throughout the course on research topics that are related to the contents of the program.
In all the planned scenarios there will be a virtual course, where different aspects of the subject are detailed.
Scenario 1: adapted normality The expository and interactive teaching will be of face-to-face character. Tutorials can be face-to-face or conducted virtually. Communication with students, in addition to face-to-face, can also be done through the virtual course forums and e-mail.
Scenario 2: distancing There will be face-to-face and virtual teaching in accordance with the coexistence formula of both modalities defined by the University of Santiago de Compostela. Synchronous virtual teaching will be done through the Microsoft Teams platform and asynchronous teaching through the Virtual Campus. In addition to doing it in person, communication with students can be done through the virtual course forums and email.
Scenario 3: closure of the facilities Teaching will be completely virtual. There will be synchronous teaching through the Microsof Teams platform and asynchronous teaching through material that complements synchronous teaching through the Virtual Campus. Communication with students will be through the virtual course forums and email.
Students will have to make expositions of some parts of the syllabus and will deliver diverse exercises. The evaluation can be completed by means of a written exam, in addition to considering the active participation in the classes and the realization of the proposed exercises.
In addition to assessing the competencies corresponding to the subject, the presentation of works aims to assess the acquisition of the competencies of the degree, with special emphasis on creativity, teamwork and the transmission of ideas.
Scenario 1: adapted normality The continuous evaluation will consist of the realization of proofs that will take place in class schedule in which each student will have to solve the exercises that indicate him. The final exam will have a theory part, which can cover definition of concepts, statement of results or total or partial proof of them. The other part will consist of solving exercises, which will be similar to those proposed throughout the course.
Scenario 2: distancing The continuous evaluation will consist of carrying out telematic tests synchronously with the virtual course tool. The final test, if it is face-to-face, will have a part of theory and another part that will consist of carrying out exercises. If it is telematic, the final test will be synchronous and will contain theoretical-practical questions and exercises.
Scenario 3: closure of the facilities The continuous evaluation will consist of the realization, in a synchronous way, of telematic tests with the tool of the virtual course. The final test will be telematic and synchronous and will contain theoretical-practical questions and exercises.
In the case of fraudulent performance of exercises or tests, the provisions of the Regulations for the evaluation of students' academic performance and review of grades will apply:
Article 16. Fraudulent performance of exercises or tests: The fraudulent performance of any exercise or test required in the assessment of a subject will involve the qualification of fail in the corresponding call, regardless of the disciplinary process that may be followed against the offending student. To be considered fraudulent, among others, the carrying out of works plagiarized or obtained from sources accessible to the public without reworking or reinterpretation and without citations to the authors and the sources.
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In accordance with the "Guidelines for the development of safe face-to-face teaching, Course 22020-2021" of the University of Santiago de Compostela, the adaptations corresponding to the sections of teaching methodology and assessment system provided for scenarios 2 and 3 are included:
Contingency plan
Teaching methodology
Scenario 2: distancing There will be face-to-face and virtual teaching in accordance with the coexistence formula of both modalities defined by the University of Santiago de Compostela. Synchronous virtual teaching will be done through the Microsoft Teams platform and asynchronous teaching through the Virtual Campus. In addition to doing it in person, communication with students can be done through the virtual course forums and email.
Scenario 3: closure of the facilities Teaching will be completely virtual. There will be synchronous teaching through the Microsof Teams platform and asynchronous teaching through material that complements synchronous teaching through the Virtual Campus. Communication with students will be through the virtual course forums and email.
Evaluation system
In the three scenarios planned as possible in this course, the final grade will be obtained from the continuous assessment, mainly based on the presentations that students must make of some parts of the syllabus. The evaluation can be completed by means of a written exam, in addition to considering the active participation in the classes and the realization of the proposed exercises.
The qualification obtained in the continuous evaluation will be applied in the two opportunities of a same academic course. If the student does not appear to the final examination (in case of being compulsory) in neither of the two opportunities will have the qualification of “Not presented”, although he has participated in the continuous evaluation.
Scenario 2: distancing The continuous evaluation will consist of carrying out, in a synchronous way, telematic tests with the Moodle tool in the virtual course. The final test, if it is face-to-face, will have a part of theory and another part that will consist of carrying out exercises. If it is telematic, the final test will be synchronous and will contain theoretical-practical questions and exercises.
Scenario 3: closure of the facilities The continuous evaluation will consist of the realization, in a synchronous way of telematic tests with the tool of the virtual course. The final test will be telematic and synchronous and will contain theoretical-practical questions and exercises.
In the case of fraudulent performance of exercises or tests, the provisions of the Regulations for the evaluation of students' academic performance and review of grades will apply:
Article 16. Fraudulent performance of exercises or tests: The fraudulent performance of any exercise or test required in the assessment of a subject will involve the qualification of fail in the corresponding call, regardless of the disciplinary process that may be followed against the offending student. To be considered fraudulent, among others, the carrying out of works plagiarized or obtained from sources accessible to the public without reworking or reinterpretation and without citations to the authors and the sources.
Eduardo Garcia Rio
Coordinador/a- Department
- Mathematics
- Area
- Geometry and Topology
- Phone
- 881813211
- eduardo.garcia.rio [at] usc.es
- Category
- Professor: University Professor
| Thursday | |||
|---|---|---|---|
| 12:00-13:00 | Grupo /CLE_01 | Spanish | Classroom 10 |
| Friday | |||
| 12:00-13:00 | Grupo /CLIL_01 | Spanish | Classroom 10 |