Differencial geometry
Code: 100107 ECTS Credits: 12| Degree | Type | Year | Semester |
|---|---|---|---|
| 2500149 Mathematics | OB | 3 | 2 |
Contact
- Name:
- Florent Balacheff
- Email:
- florent.balacheff@uab.cat
Use of Languages
- Principal working language:
- catalan (cat)
- Some groups entirely in English:
- No
- Some groups entirely in Catalan:
- Yes
- Some groups entirely in Spanish:
- No
Teachers
- Marcel Nicolau Reig
- Gil Solanes Farres
- David Marín Pérez
Prerequisites
To assimilate the contents of the subject, it is convenient to have a previous knowledge of calculus in several variables (derivation, integration, implicit function theorem), differential equations (theorem of existence and uniqueness of solutions), algebra and linear geometry (diagonalization of self-adjoint endomorphisms, quadratic forms) and topology (homeomorphism, index of a flat curve, Euler characteristic).
Objectives and Contextualisation
The concepts and notions of differential geometry and vector calculus are basic for understanding the physical reality that surrounds us. Their technical applications are also important in the field of engineering, where the objects of study can be represented geometrically by non-linear elements of the three-dimensional space, that is, basically by curves and surfaces.
The main objective is to study the geometric notions that allow theoretical characterization of the shape of these elements (curvature and torsion in the case of a curve, first and second fundamental form in the case of a surface), as well as to develop methods to calculate their metric characteristics (length, area, etc.). It is also important to relate the invariants associated with a curve contained on a surface with the notions and magnitudes of the latter. All these topics will be dealt with in the first two blocks of the course.
In the third block, the classic notions of the vector calculus will be introduced: vector fields and their line and surface integrals, as well as the integral theorems of Green, Gauss and Stokes that relate them. These results will be obtained as particular cases of the Stokes theorem for differential forms. Numerous applications of these theorems will be presented, both in physics and geometry.
Competences
- Actively demonstrate high concern for quality when defending or presenting the conclusions of one's work.
- Apply critical spirit and thoroughness to validate or reject both one's own arguments and those of others.
- Assimilate the definition of new mathematical objects, relate them with other contents and deduce their properties.
- Demonstrate a high capacity for abstraction.
- Identify the essential ideas of the demonstrations of certain basic theorems and know how to adapt them to obtain other results.
- Recognise the presence of Mathematics in other disciplines.
- Students must be capable of applying their knowledge to their work or vocation in a professional way and they should have building arguments and problem resolution skills within their area of study.
- Students must be capable of communicating information, ideas, problems and solutions to both specialised and non-specialised audiences.
- Students must develop the necessary learning skills to undertake further training with a high degree of autonomy.
- Students must have and understand knowledge of an area of study built on the basis of general secondary education, and while it relies on some advanced textbooks it also includes some aspects coming from the forefront of its field of study.
- Use computer applications for statistical analysis, numeric and symbolic calculus, graphic display, optimisation or other purposes to experiment with Mathematics and solve problems.
Learning Outcomes
- Actively demonstrate high concern for quality when defending or presenting the conclusions of one's work.
- Apply critical spirit and thoroughness to validate or reject both one's own arguments and those of others.
- Apply the integrals of line and surface to recognise some global properties of crooked and surfaces.
- Know pose and resolve curvilinear and integral integrals of surface.
- Recognise the nature of the points of an R3 curve Calculate curvature and torsion.
- Recognise the nature of the points of an R3 surface. Calculate Gauss curvature, mean curvature and principal curvatures.
- Students must be capable of applying their knowledge to their work or vocation in a professional way and they should have building arguments and problem resolution skills within their area of study.
- Students must be capable of communicating information, ideas, problems and solutions to both specialised and non-specialised audiences.
- Students must develop the necessary learning skills to undertake further training with a high degree of autonomy.
- Students must have and understand knowledge of an area of study built on the basis of general secondary education, and while it relies on some advanced textbooks it also includes some aspects coming from the forefront of its field of study.
- Topologically recognise compact spaces and their classification.
- Understand the applications of vector calculus and differential geometry calculus to physics problems.
- Use some kind of scientific program to make computations and display surfaces.
Content
1. Curves
Parameterization and length.
Curvature and torsion. Frenet Formulas
Osculating sphere.
2. Surfaces.
Regular surfaces.
First fundamental form.
The Gauss map and the second fundamental form.
Normal curvature and geodesic curvature.
Intrinsic geometry of surfaces. Gauss Egregium theorem.
Geodesics
The default theorem.
3. Vector calculus.
Differential forms.
Varieties and varieties with boundary.
Integration
Stokes theorem and applications.
The Gauss-Bonnet theorem.
Methodology
The course consists of:
1. Theory classes, where the fundamental concepts, illustrated with abundant examples, will be introduced, and the program's themes will be explained. Students will be encouraged to actively participate in class.
2. Problem solving classes, where significant exercises will be solved and certain questions that clarify the notions introduced into the theory classes will be analyzed in detail. The heuristic methods of solving mathematical problems that are characteristic of the subject will be emphasized.
3. Seminars, where the students, under the guidance of the professor, will develop and deepen certain topics that complement the subjects explained in the theoretical classes.
Annotation: Within the schedule set by the centre or degree programme, 15 minutes of one class will be reserved for students to evaluate their lecturers and their courses or modules through questionnaires.
Activities
| Title | Hours | ECTS | Learning Outcomes |
|---|---|---|---|
| Type: Directed | |||
| Problem solving classes | 30 | 1.2 | 3, 12, 9, 7, 5, 6, 11, 4, 13 |
| Theory classes | 45 | 1.8 | 3, 12, 9, 7, 5, 6, 11, 4, 13 |
| Tutoring | 15 | 0.6 | 3, 12, 9, 7, 5, 6, 11, 4, 13 |
| Type: Supervised | |||
| Seminars | 28 | 1.12 | 3, 12, 9, 7, 5, 6, 11, 4, 13 |
| Type: Autonomous | |||
| Autonomous study | 160 | 6.4 | 3, 12, 9, 7, 5, 6, 11, 4, 13 |
Assessment
The mark of the continuous evaluation of the subject, AC, will be obtained from:
1. the marks of two partial exams, E1 and E2,
2. the mark of the delivery of exercises (from the classes of problems or seminars), P,
according to the formula: AC = 0.30 E1 + 0.45 E2 + 0.25 P.
The student passes the subject if AC is greater than or equal to 5. Otherwise, the student has a recovery examination giving rise to a mark ER. The recovery examination can be used to improve the grade but notice that, in all the cases, ER will replace the sum of the two partial examination marks, E1 + E2 . The mark P of delivery of exercises is NOT recoverable and therefore the mark ER of the recovery examination will have a weight of 75% in the final grade. To be able to attend the recovery exam, the student must have been previously evaluated of continuous assessment activities that are equivalent to 2/3 of the total.
It is considered that the student presents himself for the evaluation of the course if he has participated in evaluation activities that exceed 50% of the total.
Assessment Activities
| Title | Weighting | Hours | ECTS | Learning Outcomes |
|---|---|---|---|---|
| Delivery of solved problems P | 25% | 10 | 0.4 | 2, 3, 12, 10, 9, 8, 7, 5, 6, 11, 4, 13 |
| Exam E1 | 30% | 4 | 0.16 | 2, 3, 12, 10, 9, 7, 5, 6, 11, 4 |
| Exam E2 | 45% | 4 | 0.16 | 2, 3, 12, 10, 9, 7, 5, 6, 11, 4 |
| Recovery exam ER | 75% | 4 | 0.16 | 3, 1, 12, 10, 9, 7, 5, 6, 11, 4 |
Bibliography
1. Manfredo P. do Carmo. Geometría diferencial de curvas y superfícies. Alianza Editorial. 1990
3. Sebastián Montiel y Antonio Ros. Curvas y superfícies. Proyecto Sur. 1998
4. Michael Spivak. Cálculo en Variedades. Ed. Reverté. 1970
5. Joan Girbau. Geometria diferencial i relativitat. Manuals de la UAB. 1993
eBooks:
2. Theodore Shifrin. Differential Geometry: A First Course in Curves and Surfaces. 2021
6. Victor A. Toponogov. Differential Geometry of Curves and Surfaces. Birkhäuser. 2006
7. Shoshichi Kobayashi. Differential Geometry of Curves and Surfaces. Springer. 2019
Software
SageMath