Single-Course English 5 ECTS

Differential Geometry

Overall Course Objectives

The aim of this course is to provide the students with fundamental tools and competences regarding the analysis of Riemannian manifolds as well as regarding a plethora of advanced applications of differential geometric methods and concepts.

Learning Objectives

  • Describe and apply local charts (and diffeomorphisms between them) for Riemannian manifolds – in particular with the aim of representing the Riemannian metrics concretely as indicatrix fields in the chart domain.
  • Describe, recognize, and apply isometries and conformal maps between Riemannian manifolds.
  • Describe and apply tangent spaces, vector fields, considered as derivations, and their integral curves, considered as deformation maps (flow maps).
  • Find and apply the Lie derivative, gradient, divergence, Hessian, and Laplace operators in Riemannian manifolds.
  • Apply the concept of a tensor to analyze multilinear maps.
  • Explain and apply the Levi-Civita connection in Riemannian manifolds.
  • Explain and apply the notion of parallel transport on surfaces and in Riemannian manifolds.
  • Determine geodesic curves and the exponential and logarithmic map on a surface and in a Riemannian manifold.
  • Explain the construction and meaning of the curvature tensor, the Ricci curvature, the sectional curvature and the scalar curvature in Riemannian manifolds.
  • Apply first and second variation of arc length to get global geometric and topological consequences of bounds on the curvature tensors.
  • Apply simple extensions of the above concepts and results to Lorentzian and Finslerian manifolds.
  • Apply differential geometric concepts and results to a wide spectrum of modern and significant modelling scenarios.

Course Content

Diffeomorphisms; tangent spaces; metric tensors; Poincaré models; isometries; Lie derivative; Killing vector fields; Levi-Civita connection; covariant differentiation; parallel transport; geodesics; helices and circles in Riemannian 3-manifolds; fundamental differential operators; the Laplace equation; Exponential map; Logarithmic map; first and second variation of the arc length functional; geodesic balls and spheres and their volumes; curvature operator; curvature tensor; sectional curvature; Ricci curvature; scalar curvature; the Einstein tensor; applications to Newtonian mechanics, general relativity, and to Finslerian anisotropic geometric phenomena.

Recommended prerequisites

01125/01237, Knowledge about fundamental concepts and results from calculus, geometry, and linear algebra is assumed as a prerequisite.

Teaching Method

Lectures and exercises.

See course in the course database.

Registration

Language

English

Duration

13 weeks

Institute

Compute

Place

DTU Lyngby Campus

Course code 01238
Course type Candidate
Semester start Week 5
Semester end Week 19
Days Thurs 8-12
Price

7.500,00 DKK

Registration