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 expository hours)
1. Orientability (4 expository hours)
Vector fields tangent and normal to a regular surface. Oriented atlas.
2.2 Orientability. Characterization of orientability of regular surfaces by means of normal vector fields. Oriented atlas.
2. Covariant derivative and parallel transport (7 expository hours)
2.1. Covariant derivative. Parallel vector fields
2.2. Parallel transport of a tangent vector along a curve.
3. Geodesics (8 expository hours)
3.1 Definition and examples of geodesic curves. Existence and Uniqueness of Geodesics.
3.2 Geodesic curvature.
4. The exponential map (6 expository hours)
4.1 Exponential map. Normal coordinates and polar geodesic coordinates. Gauss lemma.
4.2 The local minimizing character of geodesics.
5. Gauss-Bonnet Theorem (12 expository hours)
5.1 The rotation index of a piecewise differentiable plane curve. Geodesic curvature in an orthogonal parametrization Triangulations and the Euler-Poincaré characteristic. Local formula of Gauss-Bonnet.
5.2 Consequences of the Gauss-Bonnet Theorem.
6. Compact surfaces in R^3. Rigidity of the sphere (3 expository hours)
6.1 Lemma of Hilbert. Theorem of Liebmann. Rigidity of the sphere Theorem.
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 (2ª ed). CSIC, Madrid, 2019.
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 set forth in the Memory of the Degree in Mathematics of the University of Santiago de Compostela (USC) will be followed.
Teaching is programmed in Expository, Interactive and Tutorial classes (in very small groups).
Expository Teaching: The lectures will be devoted to the presentation and development of the essential contents of the subject.
Interactive Teaching: Interactive classes will be devoted to presenting examples and solving problems (both theoretical and the scope of applications). Individual or group work will be organized and problems will be proposed to be solved by students. In the interactive teaching classes, 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 competences.
Tutorials: The tutorial sessions are designed specially to stimulate the activity of the students outside the classroom. These will be used so that interested students can continuously examine their learning process, and for teachers to do a direct follow-up of this learning, which will allow them to detect insufficiencies and difficulties that can be corrected as they occur.
The weekly distribution of the subject will be approximately the following: 3 hours of expository class, 1 hour of interactive class. Throughout the course there will be 2 hours of tutorials (in very small groups)
The expository and interactive teaching will be essentially face-to-face, always in accordance with the formula defined by the Faculty of Mathematics. The 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 the email or through the Microsoft Teams platform.
There will be a virtual course, where all the theoretical aspects of the subject and resolved exercises appear.
CONTINGENCY PLAN in the OBSERVATIONS sectio
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.
Continuous assessment (25%). The continuous evaluation will be carried out through a written control on practical, theoretical or applicability aspects of the subject concepts.
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. The qualification obtained in the continuous assessment will be applied in the two opportunities of the same academic year (second semester and July).
Final exam (75%). A final individual (face-to-face) 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 capacity of its application to specific cases.
The written exam will consist of a part of pure theory (between 25 and 40% of the total of the test), which may include definition of concepts, statements 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. 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, the one included in the 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.
CONTINGENCY PLAN in the OBSERVATIONS section
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
Professor Elena Vázquez Abal teaches in Galician.
According to the "Guidelines for the development of safe face-to-face teaching, Course 2020-2021" of the University of Santiago de Compostela, include the adaptations corresponding to the sections of the teaching methodology and the assessment system foreseen for scenarios 2 and 3:
Contingency plan
Teaching methodology
Scenario 2: distancing
The expository and interactive teaching will be, face-to-face and virtual according to the formula of coexistence of both modalities that the Faculty of Mathematics defines. 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, email or through the Microsoft Teams platform.
Scenario 3: closure of facilities
Teaching will be completely virtual. There will be synchronous teaching through the Microsoft Teams platform and asynchronous teaching (using material that complements synchronous teaching) through the Virtual Campus. Communication with students can be done through the virtual course forums, email or through the Microsoft Teams platform.
In all the planned scenarios there will be a virtual course, where all the theoretical aspects of the subject and resolved exercises appear.
Assessment system
Scenario 2: distancing
- Continuous evaluation (30%). The continuous evaluation will be carried out through:
• 2 individual tasks written through the Virtual Campus. (90%)
• Attendance, participation of students in the classroom and tutorials (both face-to-face and virtual on the Microsoft Teams platform or in the forums of the Virtual Campus). (10%)
- Final exam (60%). Face-to-face or telematic according to the coexistence formula of both modalities defined by the Faculty of Mathematics.
Scenario 3: closure of facilities
- Continuous evaluation (40%). The continuous evaluation will be carried out through:
• 2 individual tasks written through the Virtual Campus. (90%)
• Student participation in tutorials and forums through the Virtual Campus and on the Microsoft Teams platform. (10%)
- Final exam (70%). Telematic through the Virtual Campus.
To pass the subject at the 1st opportunity, it will be essential to take the final exam and obtain a total of 5 points on average, with a minimum of 40% in the final exam.
To pass the subject in the 2nd opportunity of the call, it will be essential to sit for the final exam and obtain a total of 5 average points. Qualification of the continuous evaluation will be considered in the 2nd opportunity, if any.
Eduardo Garcia Rio
Coordinador/a- Department
- Mathematics
- Area
- Geometry and Topology
- Phone
- 881813211
- eduardo.garcia.rio [at] usc.es
- Category
- Professor: University Professor
Jose Carlos Diaz Ramos
- Department
- Mathematics
- Area
- Geometry and Topology
- Phone
- 881813363
- josecarlos.diaz [at] usc.es
- Category
- Professor: University Professor
Miguel Dominguez Vazquez
- Department
- Mathematics
- Area
- Geometry and Topology
- Phone
- 881813156
- miguel.dominguez [at] usc.es
- Category
- Researcher: Ramón y Cajal
María Ferreiro Subrido
- Department
- Mathematics
- Area
- Geometry and Topology
- maria.ferreiro.subrido [at] rai.usc.es
- Category
- Ministry Pre-doctoral Contract
Angel Cidre Diaz
- Department
- Mathematics
- Area
- Geometry and Topology
- angel.cidre.diaz [at] usc.es
- Category
- Ministry Pre-doctoral Contract
| Monday | |||
|---|---|---|---|
| 12:00-13:00 | Grupo /CLE_01 | Spanish | Classroom 02 |
| 13:00-14:00 | Grupo /CLE_02 | Galician | Classroom 06 |
| Tuesday | |||
| 12:00-13:00 | Grupo /CLE_01 | Spanish | Classroom 02 |
| 13:00-14:00 | Grupo /CLE_02 | Galician | Classroom 06 |
| Wednesday | |||
| 12:00-13:00 | Grupo /CLE_01 | Spanish | Classroom 03 |
| 13:00-14:00 | Grupo /CLE_02 | Galician | Classroom 02 |
| Thursday | |||
| 10:00-11:00 | Grupo /CLIL_01 | Galician, Spanish | Classroom 06 |
| 11:00-12:00 | Grupo /CLIL_05 | Galician | Classroom 06 |
| 11:00-12:00 | Grupo /CLIL_03 | Galician, Spanish | Classroom 08 |
| 12:00-13:00 | Grupo /CLIL_06 | Galician | Classroom 03 |
| 13:00-14:00 | Grupo /CLIL_04 | Galician | Classroom 06 |
| Friday | |||
| 12:00-13:00 | Grupo /CLIL_02 | Spanish, Galician | Classroom 08 |
| 06.05.2023 16:00-20:00 | Grupo /CLE_01 | Classroom 06 |
| 07.07.2023 16:00-20:00 | Grupo /CLE_01 | Classroom 06 |