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)
- To know the foundations of Riemannian geometry as a natural generalization of the study of surfaces in Euclidean space, distinguishing between local and global aspects of the theory, and understanding the connection with topological and analytical aspects.
- Getting students to focus, not only on the specific contents, but also on the associated methods, so that they acquire a degree of scientific maturity that allows them to face the resolution of different problems, thus awakening their ability to apply general theories to specific situations, synthesizing partial results and deducing more global ones.
- Riemannian and semi-Riemannian metrics.
- Levi-Civita connection.
- Geodesics and distance.
- Curvature: curvature tensor, sectional curvature, Ricci and scalar curvature.
- Jacobi fields: conjugate points.
- Isometric immersions and fundamental equations of submanifolds.
- Completeness: Hopf–Rinow theorem.
- Complete manifolds of negative curvature: Cartan-Hadamard theorem.
- Space forms and Cartan theorem.
- Complete manifolds of positive curvature: Myers and Synge theorems.
Basic bibliography:
- Lee, J. M.: Riemannian geometry, an introduction to curvature. Graduate Texts in Mathematics, 176. Springer-Verlag, New York (1997).
- Do Carmo, M. P.: Geometria Riemanniana. Projeto Euclides, IMPA, Rio de Janeiro (1979).
Complementary bibliography:
- Boothby, W. M.: An introduction to differentiable manifolds and Riemannian geometry. Pure Appl. Math., 120. Academic Press, Florida (1986).
- Chavel, I.: Riemannian geometry, a modern introduction. Cambridge Tracts in Mathematics, 108. Cambridge University Press, Cambridge (1993).
- Lee, J. M.: Introduction to Riemannian Manifolds. Second Edition. Graduate Texts in Mathematics, 176. Springer-Verlag (2018).
- O'Neill, B: Semi-Riemannian Geometry with applications to relativity. Pure Appl. Math., 103. Academic Press, New York-London (1983).
- Petersen, P.: Riemannian geometry. Third edition. Graduate Texts in Mathematics, 171, Springer, Cham (2016).
- Sakai, T.: Riemannian geometry. Transactions of Mathematical Monographs 149, American Mathematical Society, Providence, RI (1996).
CB6 - Possessing and understanding knowledge that provides a foundation or opportunity to be original in the development and/or application of ideas, often in a research context.
CB7 - Let students know how to apply acquired knowledge and their ability to solve problems in new or little-known situations within broader (or multidisciplinary) contexts related to their area of study.
CB8 - That students are able to integrate knowledge and face the complexity of formulating judgments from information that, being incomplete or limited, includes reflections on social and ethical responsibilities linked to the application of their knowledge and judgments.
CB9 - Let students know how to communicate their conclusions and the ultimate knowledge and reasons that sustain them to specialized and non-specialized audiences in a clear and unambiguous way.
CB10 - That students possess the learning skills that allow them to continue studying in a way that will have to be largely self-directed or autonomous.
CG01 - Introducing to research to students, as an integral part of a deep training, preparing them for the eventual completion of a doctoral degree.
CG02 - Acquisition of high-level mathematical tools for various applications covering the expectations of graduates in mathematics and other basic sciences.
CG03 – Know the broad picture of current mathematics, both in its lines of research, and in methodologies, resources and problems it addresses in various fields.
CG04 - Train for analysis, formulation and problem solving in new or little-known environments, within broader contexts.
CG05 - Prepare for decision-making from abstract considerations, to organize and plan and to solve complex issues.
CT01 - Use bibliography and search tools for general and math-specific bibliographic resources, including Internet access.
CT02 - Optimally manage working time and organize available resources, setting priorities, alternative paths, and identifying logical errors natoma of decisions.
CT03 – Enhancing capacity for work in cooperative and multidisciplinary environments.
CE01 - Training for study and research in developing mathematical theories.
CE02 - Apply the tools of mathematics in various fields of science, technology and social sciences.
CE03 – Develop the necessary skills for the transmission of mathematics, oral and written, both in terms of formal correction, and in terms of effectiveness, emphasizing the use of appropriate ICT.
We will follow the general directives established in the Master in Mathematics of the University of Santiago de Compostela (USC).
A key aspect in teaching at any educational level is the motivation of the concepts that are introduced. Thus, in Mathematics it is necessary to adopt a methodological approach that first introduces the notions and results that will be studied by means of examples. In this initial phase, the new concepts must be connected in a natural way with previous knowledge, in order to contribute to a unifying image of Mathematics. After this first stage, the properties, results and methods associated with the concepts will be developed. Finally, such contents will be reinforced through more examples, exercises and problems of different difficulty and nature. In addition, in accordance with the spirit of the European Higher Education Area, where students become an active subject and motor of their own learning, many of these exercises and problems must be carried out by the students, in order to consolidate and assimilate contents.
Among the teaching methodologies presented in the syllabus, we will use, above all:
M1 Teacher expositions
M2 Presentations by the students
M3 Resolution of exercises
M4 Reading and study by students
M5 Discussions in class
M9 Summaries and proposed works
M10 Complementary readings
Each student's grade will be achieved through continuous assessment and completion of a final test.
Continuous evaluation can be carried out through controls, work delivered and participation of the student in the classroom. Each student's grade will not be lower than that of the final test or the one obtained by weighting it with the continuous assessment, giving the latter a weight of 25%.
At the first opportunity, the final test will consist of the delivery of a small written work and presentation in the classroom of some aspect of the subject (theorem, examples, constructions, etc.) assigned by the teacher to each student. Alternatively, each student may choose to take a written final exam. At the second opportunity, the final test will consist of the completion of a written examination, and the grade of the continuous assessment obtained will be preserved.
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.
In-person work in the classroom: 24 hours
Personal work outside the classroom: 51 hours
Total volume of work: 75 hours
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Miguel Dominguez Vazquez
Coordinador/a- Department
- Mathematics
- Area
- Geometry and Topology
- Phone
- 881813156
- miguel.dominguez [at] usc.es
- Category
- Professor: University Lecturer
| Monday | |||
|---|---|---|---|
| 11:00-12:00 | Grupo /CLE_01 | Galician | Classroom 10 |
| Friday | |||
| 11:00-12:00 | Grupo /CLIL_01 | Galician | Classroom 10 |
| 01.15.2025 10:00-14:00 | Grupo /CLE_01 | Classroom 10 |
| 06.13.2025 10:00-14:00 | Grupo /CLE_01 | Classroom 10 |