Differential Geometry

COURSE OBJECTIVE

• The student understands how to read and write in the language of coordinate free analysis and geometry.
• The student can reproduce the most important arguments and constructions in differential geometry.
• The student can apply these to compute on concrete geometric objects (manifolds).
• The student can translate between geometric intuition and mathematical statements.

COURSE CONTENT
This course is an introduction to the theory of manifolds. These may be interpreted as generalisations of curves and surfaces to arbitrary dimensions. Apart from giving the most relevant definitions from differential topology (manifolds, vector bundles and differential forms), we make short excursions to Riemannian geometry (Riemannian metric), dynamical systems (flows of vector fields) as well as to algebraic topology (fundamental group and de Rham cohomology).

More precisely, the subject list includes:

• Submanifolds, manifolds, tangent vectors
• Smooth maps (between manifolds), differential,immersions/submersions/embeddings
• Vector fields and their flows, Lie bracket, Lie groups
• Vector bundles, tangent bundles, tensor products, sections
• Riemannian metrics, distances on Riemannian manifolds
• Differential forms, pullbacks, exterior derivative
• Stokes’ Theorem and De Rham cohomology
• Fundamental group and outlook towards algebraic topology

TEACHING METHODS
Lectures and tutorials

TYPE OF ASSESSMENT
Homework (makes up for 30% of the final grade), written final exam

RECOMMENDED BACKGROUND KNOWLEDGE
Analysis 1, Linear Algebra 1, Analysis 2 and 3, Topology

REMARKS
In order to compensate for the fact that we will not strictly follow one book, there will be lecture notes made available.