Course content

The aim of the course is to introduce fundamental concepts and examples in differential geometry. Key concepts that will be discussed include differentiable structures and smooth manifolds, tangent bundles, embeddings, immersions and submersions and regular points. Important examples of manifolds such as surfaces, spheres, and manifolds with boundary are discussed. Beyond key concepts a selection of topics from Riemannian geometry, connections, symplectic geometry or the theory of differential forms will be studied. Applications presented in the course may range from flows on manifolds and geodesics, to Lie groups, integration on manifolds and Stokes’ theorem. These methods and ideas have been influential to and are used in many other parts of mathematics and physics as well as in other areas of application.

Learning outcome

1. Knowledge: The student has knowledge of fundamental concepts and methods in differential geometry, and of examples of manifolds.

2. Skills: The student is able to apply his or her knowledge of differential geometry to formulate and solve problems of a geometrical nature in mathematics.

Learning methods and activities

The learning methods and activities depend on the course teacher, but will in general consist of lectures and exercises. The course will be given in autumn in years of odd numbers.

Compulsory assignments

• Works

Recommended previous knowledge

The course is based on TMA4401/13 Mathematics 1D/2D and/or MA1101 Basic Calculus I, MA1102 Basic Calculus II, MA1103 Vector Calculus, MA1201 Linear Algebra and Geometry and MA1202 Linear Algebra with Applications or equivalent. We recommend to take the course as a follow-up or together with TMA4190 Introduction to Topology.

Course materials

Will be announced at the start of the course.