# Differential Geometry and Vector Bundles

Spring 2010

Lecturer: Dr Theodore Voronov (Alan Turing 2.109). Classes:
Tuesday 13.00-14.50 (two hours), Frank Adams Room 1 (Alan Turing 1.212)

The 3rd and 4th year students are most welcome to attend the course. Since there is no formal assessment, a possible way of receiving formal credit for it can be in a form of "course project" (details to be discussed).

I wish to give an introduction to the main notions of modern differential geometry such as connection (and curvature) in the context of vector bundles, and also touch on topology of vector bundles. A rough explanation of these concepts is as follows.

"Fiber bundles" ("vector bundles" in particular) play fundamental role in geometry/ topology; in physics they arise as natural language for describing physical fields. A fiber bundle is a manifold (or a topological space) that locally looks like the product of one space called the base with another space called the (standard) fiber. The whole space is the union of copies of the fiber parametrized by the 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 x I. Particularly important are vector bundles, for which fibers are vector spaces. An example of a vector bundle is the tangent bundle of a manifold.

Maps taking points of the base to fibers over these points are called sections of fiber bundles. In particular, for vector bundles, sections can be added and multiplied by functions. (Algebraically, they make a module over the algebra of functions on the base, which turns out to be a projective module. Actually, it is the best example of a projective module which is not necessarily free.) However, sections cannot be differentiated naturally so that the derivative of a section be again a section. For example, sections of the tangent bundle are vector fields on a manifold. Differentiating vector fields or sections of more general vector bundles requires introducing an extra structure known as a connection or covariant derivative. Now, the curvature of a connection is a new phenomenon which does not occur for ordinary derivatives. Its simplest example is the 'internal curvature' of a 2-surface in Euclidean 3-space, which is responsible for the fact that it is impossible to map a region of a sphere onto 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, which may also be seen as the simplest manifestation (and prototype) of the celebrated Atiyah-Singer index theorem.

Textbooks:

No single textbook is followed. There are many good sources on differential geometry on various levels and concerned with various parts of the subject. Some examples are listed below. More books 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.

Assessment:
No formal assessment, but undergraduate students wishing to attend and get credit may speak with the lecturer about counting it as a course project.

### Tentative topics:

(I may change things on the way)

1. Vector bundles.
Idea of a fiber bundle. Non-trivial bundles. Formal definition of a fiber bundle. Transition functions and the cocycle property. Vector bundles. Sections. Local frames.

2. Connection on a vector bundle.
Axioms of a covariant derivative. Local connection 1-form. Existence of connection. Connection defined by a projector.

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

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

5. Curvature of a connection.
Theorema Egregium for surfaces and its meaning. Definition of the curvature 2-form for a connection on a vector bundle. Properties and interpretations.

6. Parallel transport.
Examples and the main idea. Equation of parallel transport. Geodesic lines.

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

8. Classification of vector bundles.
Pull-back of vector bundles. Homotopy property. Embedding into a trivial bundle. Classifying spaces.

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

Theodore Voronov 19 January (1 February) 2010