Course Code : | 1001WETDIM |

Study domain: | Mathematics |

Academic year: | 2019-2020 |

Semester: | 1st semester |

Sequentiality: | - |

Contact hours: | 60 |

Credits: | 6 |

Study load (hours): | 168 |

Contract restrictions: | No contract restriction |

Language of instruction: | Dutch |

Exam period: | exam in the 1st semester |

Lecturer(s) | Tom Mestdag Jens Hemelaer |

At the start of this course the student should have acquired the following competences:

an active knowledge of

- competences corresponding the final attainment level of secondary school

an active knowledge of

- Dutch

The students have a solid background in elementary analysis and linear algebra. They know the basics of differential and integral calculus, vector spaces, linear transformations and euclidean spaces.

- The students can work with curves and surfaces, and with the invariants that characterize them locally (torsion and curvature of curves, and geodesic, normal and Gauss curvature for surfaces). Through the concept of a manifold, they understand the difference between a local and a global geometric property.

1. Curves

Curves in 3-dimensional Euclidean space. The canonical representation of a curve. The fundamental theorem.

2. Surfaces

Surfaces in 3-dimensional Euclidean space. Tangent plane. The first fundamental form. Isometries. Normal and geodesic curvature. Weingarten map, Gaussian curvature, mean curvature, principal curvatures. Christoffel symbols, geodesic lines, the Gauss-Bonnet formula, Riemann curvature tensor. Theorema Egregium.

3. Riemannian manifolds

Generalities about differential manifolds. Examples: matrix groups and surfaces in R^3. Tangent spaces of a manifold. Riemannian manifolds. Parallel transport and linear connections. The Jacobi equation and the Riemann curvature tensor. Other areas of modern differential geometry.

Class contact teachingLectures Practice sessions

Personal workExercises Assignments Individually Assignments In group

Personal work

ExaminationWritten examination without oral presentation Closed book Open book Open-question

The course notes can be downloaded from blackboard for free.

M.P. Do Carmo, Differential Geometry of Curves and Surfaces, Prentice Hall (1976)

T.J. Willmore: Riemannian Geometry, Clarendon Press (Oxford) (1993).

S. Sternberg, Curvature in Mathematics and Physics, Dover Publications (2012)

Tom Mestdag

tom.mestdag@uantwerpen.be