ECTS credits ECTS credits: 6
ECTS Hours Rules/Memories Student's work ECTS: 99 Hours of tutorials: 3 Expository Class: 24 Interactive Classroom: 24 Total: 150
Use languages Spanish, Galician
Type: Ordinary Degree Subject RD 1393/2007 - 822/2021
Departments: Mathematics
Areas: Geometry and Topology
Center Faculty of Mathematics
Call: Second Semester
Teaching: With teaching
Enrolment: Enrollable
To use differential and integral calculus along with Euclidean topology for the study of curves and surfaces in the 3-dimensional euclidean space. To show how to apply differential equations and line and surface integrals to determine global properties of curves and surfaces. To work with tangent and normal vector fields to a surface and to understand the parallel transport of vectors along curves on surfaces. To know how to recognize the geodesics on the surfaces. To assimilate the most outstanding properties and theorems of global differential geometry of surfaces, including orientability and the Gauss-Bonnet theorem.
0. Review of basic notions of curves and regular surfaces (2 hours)
1. Orientability (4 hours)
1.1 Tangent and normal vector fields to a regular surface.
2.2 Orientability. Oriented atlas. Characterization of orientability of regular surfaces. Oriented bases.
2. Covariant derivative and geodesics (12 hours)
2.1. Covariant derivative. Parallel vector fields.
2.2 Geodesics: definition and examples. Existence and Uniqueness of Geodesics.
2.3 Geodesic curvature.
2.4. Parallel transport of a tangent vector along a curve.
3. The exponential map (9 hours)
3.1 Exponential map. Normal coordinates and polar geodesic coordinates. Gauss lemma.
3.2 The minimizing property of geodesics.
4. Gauss-Bonnet Theorem (12 hours)
4.1 The rotation angle of a piecewise differentiable plane curve. Geodesic curvature in an orthogonal parametrization. Local Gauss-Bonnet theorem.
4.2 Triangulations and the Euler-Poincaré characteristic. Global Gauss-Bonnet theorem.
4.2 Consequences of the Gauss-Bonnet Theorem.
5. Compact surfaces in R^3. Rigidity of the sphere (3 hours)
5.1 Lemma of Hilbert. Theorem of Liebmann. Rigidity of the sphere.
Basic bibliography
CARMO do, Manfredo Perdigão. Differential Geometry of Curves and Surfaces. Prentice Hall. Englewood Cliffs, 1976.
HERNÁNDEZ CIFRE, María de los Ángeles & PASTOR GONZÁLEZ, José Antonio. Un curso de Geometría Diferencial. CSIC, Madrid, 2010.
Complementary bibliography
ARAÚJO, Paulo Ventura. Geometria Diferencial. Coleçao Matemática Universitaria. IMPA, Río de Janeiro 1998.
ABATE, Marco & TOVENA, Francesca. Curves and Surfaces. Springer-Verlang Italia, 2012.
ABBENA, Elsa; GRAY, Alfred & SALAMON, Simon. Modern Differential Geometry of Curves and Surfaces with Mathematica, Third Edition. Taylor & Francis Group 2006.
MONTIEL, Sebastián & ROS Antonio. Curvas y superficies, Proyecto Sur D.L., Granada, 1998.
RODRÍGUEZ-SANJURJO, José Manuel & RUÍZ, Jesús María. Introducción a la Geometría Diferencial II: Superficies. Ed. Sanz y Torres, 2019
GENERAL COMPETENCES
CX1.- To learn the most important concepts, methods and results of the different branches of mathematics, together with some historical perspective of their development.
CX3.- To apply practical and theoretical mathematical knowledge, as well as analytical and abstraction capabilities, in posing and solving problems in any professional or academic context.
CX4.- To communicate in oral and written form mathematical knowledge, ideas, methods and results to either specialists or non-specialists in the field.
CX5.- To study and learn autonomously, with good organization in the use of time and resources, new results and techniques in any scientific or technological discipline.
SPECIFIC COMPETENCES
CE1.- To understand and use the mathematical language.
CE2.- To understand the rigorous proofs of some classical mathematical theorems.
CE3.- To be able to provide proofs of mathematical results. To formulate conjectures, and to design strategies to prove them or to show that they are false.
CE4.- To identify mistakes or wrong arguments, and provide proofs or counterexamples.
CE5.- To efficiently grasp the content or definition of a new mathematical object, and be able to relate it/them with known objects and to use it/them in different contexts.
CE6.- To be able to identify and make abstractions of the fundamental issues of a given problem, and be able to distinguish them from those that are purely circumstantial.
TRANSVERSAL COMPETENCES
CT1.- To use appropriate references as well as searching tools and bibliographic mathematical resources, including internet browsing.
CT2.- To use in optimal form the working time and to organize the available resources by establishing priorities and alternative approaches, as well as identifying logic errors in any decision making process.
CT3.- To corroborate or to refute in a rational way others’ arguments.
CT4.- To be able to read science, both in mother tongue as well as in foreign language when the latter is relevant for the scientific issue under study; specially in English.
The general methodological indications of the Degree in Mathematics of the University of Santiago de Compostela (USC) will be followed.
Teaching is programmed in Expository, Interactive and Tutorial lectures.
Expository Teaching: Lectures will be devoted to the presentation and development of the essential contents of the subject.
Interactive Teaching: Interactive lectures will be devoted to presenting examples and solving problems (theory and applications). Individual or team work will be organized and problems will be proposed to be solved by students. In the interactive teaching lectures, maximum participation and implication of the students is required, as the discussion, debate and resolution of the proposed tasks, aim to practice and strengthen their knowledge and to work some of the aforementioned competencies.
Tutorials: Tutorial sessions are designed specially to stimulate the activity of the students outside lectures. Interested students can continuously examine their learning process, and teachers will directly follow their learning, allowing to detect insufficiencies and difficulties that can be corrected as they occur.
The weekly distribution of the subject will be approximately as follows: 3 hours of expository lectures, 1 hour of interactive lectures. Throughout the course there will be 2 hours of tutorials (in very small groups).
Expository and interactive teaching will be essentially face-to-face, always in accordance with the formula defined by the Faculty of Mathematics. Tutorials and communication with the students can be face-to-face or take place virtually. In the virtual case it will be possible to do it through the forums of the virtual course, email, or through the Microsoft Teams platform.
There will be a virtual course, where theoretical aspects of the subject and solved exercises appear.
Without prejudice to the general evaluation criteria for all subjects of the Degree, for the calculation of the final grade, the qualification of the continuous assessment and the qualification of the final exam will be considered.
The continuous assessment (25%) will consist of a test that will be carried out in class, in which each student will have to solve previously proposed exercises. In addition, participation in expositive and interactive lectures and tutorials will be considered within the continuous assessment. The qualification obtained in the continuous assessment will be applied in the two opportunities during the same academic year (second semester and last). If the student does not sit the final exam in neither of the two opportunities, they will have the qualification of "Not presented", even if they have participated in the continuous assessment.
Final exam (75%). A final written exam will be carried out, which will allow to verify the knowledge acquired in relation to the concepts and results of the subject and the capability to apply it to specific cases. The final exam will consist of a purely theoretical part, which may include definition of concepts, statement of results, or total or partial proof of them, and another part that will be the resolution of exercises, which will be similar to the ones proposed during the semester. Each of these parts (theory and exercises) will have a weight between 40% and 60% of the total qualification.
The evaluation system (continuous assessment and written exam) will be equivalent for both groups, but not necessarily the same. The grading obtained by the student will not be less than the grading obtained in the final exam.
Through the different proposed activities, of course, by contextualizing the subject in 3rd grade, the acquisition of competences, such as CX3, CX4, CX5, CE1, CE3, CE4, CE5, CE6, CT1, CT2, CT3 and CT5, or he ability to work in a team and the autonomous learning will be evaluated.
In addition to the specific competences of the subject, the competences CX1, CX3, CX4, CE1, CE2, CE3, CE4, CE5, CE6 and CE6 will be evaluated.
In the case of fraudulent performance of exercises or tests, Regulations for the evaluation of the academic performance of students and review of grades will be applied:
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 a failure grade in the corresponding call, regardless of the disciplinary process that may be followed against the offending student. Considering fraudulent, among others, the carrying out of plagiarized works or obtained from sources accessible to the public without reworking or reinterpretation and without quotes to the authors and sources.
Time in presence of the teacher:
Expository classes: 42 h.
Interactive laboratory classes: 14 h.
Tutorials in very small groups or individualized: 2 h.
Total hours of classroom work in class 58
Time of personal work:
Self study, individual or group: 55 h.
Writing of exercises, conclusions and other works: 27 h.
Programming / experimentation or other work in computer / laboratory: 5 h.
Suggested readings, activities in a library or similar: 5h.
Total hours of personal work of the students 92
Total volume of work: 150 hours
Prerequisites to follow this course are the following courses:
- Linear and Multilinear Algebra
- Topology
- Diferentiation of Real Functions of Several Variables
- Integration of Real Functions of Several Variables
- Introduction to Ordinary Differential Equations
- Curves and Surfaces
María Elena Vázquez Abal
- Department
- Mathematics
- Area
- Geometry and Topology
- Phone
- 881813143
- elena.vazquez.abal [at] usc.es
- Category
- Professor: University Professor
Jose Carlos Diaz Ramos
Coordinador/a- Department
- Mathematics
- Area
- Geometry and Topology
- Phone
- 881813363
- josecarlos.diaz [at] usc.es
- Category
- Professor: University Professor
| Tuesday | |||
|---|---|---|---|
| 11:00-12:00 | Grupo /CLE_01 | Galician | Classroom 03 |
| 13:00-14:00 | Grupo /CLE_02 | Galician | Classroom 06 |
| Wednesday | |||
| 12:00-13:00 | Grupo /CLE_01 | Galician | Classroom 03 |
| 13:00-14:00 | Grupo /CLE_02 | Galician | Classroom 02 |
| 13:00-14:00 | Grupo /CLIL_01 | Galician | Classroom 03 |
| Thursday | |||
| 11:00-12:00 | Grupo /CLIL_05 | Galician | Classroom 06 |
| 12:00-13:00 | Grupo /CLIL_06 | Galician | Classroom 03 |
| 12:00-13:00 | Grupo /CLIL_02 | Galician | Classroom 09 |
| 13:00-14:00 | Grupo /CLIL_04 | Galician | Classroom 06 |
| 13:00-14:00 | Grupo /CLIL_03 | Galician | Classroom 09 |
| 05.19.2025 16:00-20:00 | Grupo /CLE_01 | Classroom 06 |
| 07.04.2025 16:00-20:00 | Grupo /CLE_01 | Classroom 06 |