Differential Geometry

MATH41122: Level 4 (15 credits)

Spring 2009

Lecturer: Dr Theodore Voronov (Alan Turing 2.109). Classes:
    Thursday 13.00-13.50, Alan Turing G.209
    Friday 15.00-16.50, Alan Turing G.108

Last updated: 3 (16) May 2009
Brief description:

This course unit introduces the main notions of modern differential geometry, such as connection and curvature. It builds on the course unit MATH31061/MATH41061 Differentiable Manifolds. A natural language for describing various 'fields' in geometry and its applications such as physics is that of fiber bundles. These are manifolds (or topological spaces) that locally look like the product of a piece of one space called the base with another space called the fiber. The whole space is the union of copies of the fiber parametrized by points of the base. A good example is the Möbius band which locally looks like the product of a piece of a circle S1 with an interval I, but globally involves a "twist", making it different from the cylinder S1× I. A 'field' (or a section) associates to each point in the base a point in the fiber attached to this point. In order to differentiate sections we need an extra structure known as a connection or covariant derivative. It often comes naturally in examples such as surfaces in Euclidean space. In this case a covariant derivative of tangent vectors can be defined as the usual derivative in the Euclidean space followed by the orthogonal projection onto the tangent plane. The curvature of a connection in a fiber bundle is a new phenomenon which does not exist for the derivative of ordinary functions. It generalizes the 'internal' curvature of a surface (which is responsible for the fact that it is impossible to map a region of a sphere onto a flat surface preserving distances). Analysis of curvature on vector bundles directly leads to their topological invariants such as characteristic classes. A prototype of such a relation for the tangent bundle of a surface is given by the classical Gauss-Bonnet theorem.

Textbooks:

No single textbook is followed. It is important to keep the lecture notes. There are many good sources on differential geometry on various levels and concerned with various parts of the subject. Below is a list of books that may be useful. More sources can be found by browsing library shelves.

B. A. Dubrovin, A. T. Fomenko, S. P. Novikov. Modern geometry, methods and applications.
A. S. Mishchenko, A. T. Fomenko. A course of differential geometry and topology.
S. Kobayashi, K. Nomizu. Foundations of differential geometry.
S. Morita. Geometry of differential forms.
R. Wells. Differential analysis on complex manifolds.

Prerequisites, co-requisites, and dependent courses:
Prerequisite: Differentiable Manifolds. Introduction to Topology may be a plus.
Dependent courses: formally none; however, differential geometry is one of the pillars of modern mathematics; its methods are used in many applications outside mathematics, including physics and engineering.

Assessment:
Final mark = 20% coursework + 80% exam.
Coursework mode: take home work; deadline: TBA.
Exam mode: 3 hour exam at the end of semester.


Topics:

1. Introduction: connections and fiber bundles.
    The problem of a 'covariant' differentiation of vector fields. Examples. Idea of a fiber bundle, in particular, a vector bundle. Non-trivial bundles.
    Problems. Lecture notes.

2. Vector bundles.
    Definition of a fiber bundle. Transition functions and the cocycle property. Vector bundles. Sections. Local frames.
    Problems. Lecture notes.

3. Connection on a vector bundle.
    Axioms of a covariant derivative. Local connection 1-form. Existence of connection. Connection defined by a projector.
    Problems. Lecture notes.

4. Metric and connection.
    Metric (scalar or inner product) on a vector bundle. Connection compatible with metric. Property of local connection 1-forms. Existence.
    Problems. Lecture notes.

5. Constructions of connections.
    Connections on manifolds. Christoffel symbols. Torsion. Levi-Civita theorem and Christoffel formulas. Holomorphic bundles over complex-anlytic manifolds and Chern theorem.
    Problems. Lecture notes.

6. Connections on surfaces and intrinsic curvature.
    Derivation formulas. Curvature of curves on a surface: "normal" and "geodesic" parts. Gaußian curvature. Theorema Egregium and its meaning.
    Problems. Lecture notes.

7. Curvature of a connection.
    Definition of the curvature 2-form for a connection on a vector bundle. Properties and interpretations.
    Problems. Lecture notes.

8. Parallel transport.
    Examples and the main idea. Equation of parallel transport. Geodesic lines.
    Problems. Lecture notes.

9. Characteristic classes.
    Example: the first Chern class. Chern--Weil construction of characteristic classes from invariant polynomials.
   Lecture notes.

10. Classification of vector bundles.
    Pull-back of vector bundles. Homotopy property. Embedding into a trivial bundle. Classifying spaces. More on characteristic classes.
    Lecture notes.

11. Characteristic classes and topological invariants.
    Curvature and parallel transport. Parallel transport over a closed contour. Rotation of a vector under the parallel transport on a surface. Excess of a geodesic triangle. Gauß-Bonnet theorem for triangulated surfaces.
    Lecture notes.

http://www.maths.manchester.ac.uk/~tv/dif_geometry.html

Theodore Voronov 3 (16) May 2009