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. (15 h.)
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. (10 h)
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.
3 Lorentzian and semi-Riemannian geometry (5 h.)
3.1. Semi-riemannian metrics: existence.
3.2. Local properties: curvature and degenerate planes.
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.
- Training for the study and research in developing mathematical theories.
- Develop the necessary skills for the transmission of mathematics, oral and written, both in terms of formal correctness, and in terms of communicative efficiency, highlighting the use of appropriate ICT.
- Use bibliography and search tools for general and specific bibliographic resources of Mathematics, including Internet access.
- Manage work time in an excellent way and organize available resources, establishing priorities, alternative paths and identifying logical errors in decision making.
- Enhance the ability to work in cooperative and multidisciplinary environments.
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.
The qualification of each student will be done through continuous evaluation and the completion of a final test.
The continuous evaluation will be carried out by means of controls, work delivered and the student's participation in the classroom. The student's grade will not be lower than that of the final test or the one obtained by weighting it with the continuous evaluation, giving the latter a weight of 25%.
In the case of fraudulent performance of exercises or tests, the provisions of the Regulation for the evaluation of the academic performance of students and the review of grades will apply:
Article 16. Fraudulent performance of exercises or tests: The fraudulent performance of any exercise or test required in the evaluation of a subject will imply the qualification of failing in the corresponding call, regardless of the disciplinary process that may be followed against the offending student. Being considered fraudulent, among others, to carry out works plagiarized or obtained from sources accessible to the public without reworking or reinterpretation and without citing the authors and sources.
The recommended dedication would be at least 30 hours of personal work.
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Eduardo Garcia Rio
Coordinador/a- Department
- Mathematics
- Area
- Geometry and Topology
- Phone
- 881813211
- eduardo.garcia.rio [at] usc.es
- Category
- Professor: University Professor
| Tuesday | |||
|---|---|---|---|
| 10:00-11:00 | Grupo /CLE_01 | Spanish | Classroom 10 |
| Thursday | |||
| 12:00-13:00 | Grupo /CLIL_01 | Spanish | Classroom 10 |
| 06.28.2023 10:00-14:00 | Grupo /CLE_01 | Classroom 10 |