Differential geometry I - curves and surfaces
Summary
This course serves as an introduction to classical differential geometry, which studies the geometric properties of curves and surfaces in the Euclidean space.
Content
- Curves in the Euclidean plane and the Euclidean space.
- The notion of submanifolds of Euclidean space, charts, local parametrizations, the tangent space.
- The metric tensor (first fundamental form) of a parametrized surface.
- Curvature of regular surfaces (second fundamental form, Gaussian curvature, mean curvature, principal curvatures).
- Isometric surfaces. Gauss's Theorema Egregium.
- Notions of hyperbolic geometry.
- A short introduction to notions of abstract manifolds.
Keywords
Curves, surfces, submanifolds, curvature, torsion, geodesics, first/second fundamental form, differential geometry, hyperbolic geometry.
Learning Prerequisites
Required courses
All first year courses in the mathematics (or physics) programme.
Learning Outcomes
By the end of the course, the student must be able to:
- Give an example of a regular curve and a regular surface and know how to parametrize them.
- State the definitions given in class.
- Prove the theorems presented in class.
- Develop a geometric intuition around the concepts presented in class.
- Develop the ability to perform geometric calculations.
- Solve the problems in the exercise sheets.
Transversal skills
- Use a work methodology appropriate to the task.
- Demonstrate a capacity for creativity.
- Demonstrate the capacity for critical thinking
Teaching methods
Weekly lectures and exercise sessions.
Expected student activities
Study and understand the concepts presented in class, solve the exercises.
Assessment methods
Written exam.
Supervision
| Office hours | No |
| Assistants | Yes |
| Forum | No |
Resources
Virtual desktop infrastructure (VDI)
No
Bibliography
The course will be based on the following books, which can be accessed via the EPFL library:
- S. Kobayashi, "Differential geometry of curves and surfaces",
- M. do Carmo, "Differential geometry of curves and surfaces".
The following books can be used as alternative resources:
- T. Needham, "Visual differential geometry and forms",
- V. Toponogov, "Differential geometry of curves and surfaces",
- M. Umehara, "Differential geometry of curves and surfaces".
Ressources en bibliothèque
Moodle Link
Prerequisite for
The rest of the courses on the differential geometry series (manifolds, Riemannian geometry, general relativity), algebraic geometry.
Dans les plans d'études
- Semestre: Automne
- Forme de l'examen: Ecrit (session d'hiver)
- Matière examinée: Differential geometry I - curves and surfaces
- Cours: 3 Heure(s) hebdo x 14 semaines
- Exercices: 3 Heure(s) hebdo x 14 semaines
- Type: obligatoire