# Differential geometry

Course Code : | 1001WETDIM |

Study domain: | Mathematics |

Academic year: | 2017-2018 |

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 |

### 1. Prerequisites *

- 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.

### 2. Learning outcomes *

- 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.

### 3. Course contents *

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.

### 4 International dimension*

### 5. Teaching method and planned learning activities

Personal work

### 6. Assessment method and criteria

### 7. Study material *

#### 7.1 Required reading

The course notes can be downloaded from blackboard for free.

**7.2 Optional reading**

The following study material can be studied voluntarily :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)

### 8. Contact information *

Tom Mestdag

tom.mestdag@uantwerpen.be