Master
2018/2019

## Mathematics of Science

Type:
Compulsory course (Mathematics)

Area of studies:
Mathematics

Delivered by:
Faculty of Mathematics

When:
1 year, 1, 2 module

Mode of studies:
Full time

Instructors:
Alexander S. Tikhomirov

Master’s programme:
Mathematics

Language:
English

ECTS credits:
5

### Course Syllabus

#### Abstract

The notion of vector, respectively, principal bundle over a smooth manifold is one of the central notions in modern mathematics and its applications to mathematical and theoretical physics. In particular, all known types of physical interations (gravitational, electromagnetic, etc.) are described in terms of connections and other geometric structures on vector/principal bundles on underlying manifolds. The properties of physical fields can be formulated in terms of geometric invariants of connections such as curvature and characteristic classes of corresponding vector/principal bundles. In this course we give an introduction to the differential geometry of vector and principal bundles and consider their metrics, connections, curvature and characteristic classes. Some applications to algebraic geometry, topology, and gauge theory of classical fields (in particular, Maxwell equations, Yang-Mills theory) are discussed.

#### Learning Objectives

- To bring together and to shape systematically the basic notions of differential geometry

#### Expected Learning Outcomes

- Show and teach the application of modern methods of mathematics to the problems of modern natural Sciences

#### Course Contents

- Manifolds.Manifolds and smooth maps. Tangent vectors. Vector fields. Differential forms. Exterior differentiation on a manifold. Exterior differentiation on R3. Pullback of differential forms.
- Riemannian Manifolds.Inner products on a vector space. Riemannian metric. Existence of a Riemannian metric. Regular curves. Arc length parametrization. Signed curvature of a plane curve. Orientation and curvature.
- Affine Connections.Affine connections. Torsion and curvature. The Riemannian connection
- Vector Bundles.Definition of a vector bundle. The vector space of sections. Extending a local section to a global section. Local operators. Restriction of a local operator to an open subset. Frames. F-linearity and bundle maps. Multilinear maps over smooth functions.
- Connections on a Vector BundleConnections on a vector bundle. Existence of a connection on a vector bundle. Curvature of a connection on a vector bundle. Riemannian bundles. Metric connections. Restricting a connection to an open subset. Connections at a point.
- Connection, curvature, and torsion formsConnection and curvature forms. Connections on a framed open set. Metric connection relative to an orthonormal frame. Connections on the tangent bundle. Covariant differentiation along a curve. Connection-preserving diffeomorphisms. Christoffel symbols.
- Geodesics.The definition of a geodesic. Reparametrization of a geodesic. Existence of geodesics. Geodesics in the Poincar´e half-plane. Parallel translation. Existence of parallel translation along a curve. Parallel translation on a Riemannian manifold
- Exponential mapsThe exponential map of a connection. The differential of the exponential map. Normal coordinates. Left-invariant vector fields on a Lie group. Exponential map for a Lie group. Naturality of the exponential map for a Lie group. Adjoint representation. The exponential map as a natural transformation.
- Distance and volume.Distance in a Riemannian manifold. Geodesic completeness. Dual 1-forms under a change of frame. The volume form in local coordinates.
- Operations on vector bundlesVector subbundles. Subbundle criterion. Quotient bundles. The pullback bundle. Examples of the pullback bundle. The direct sum of vector bundles. Other operations on vector bundle.
- Vector-valued forms.Vector-valued forms as sections of a vector bundle. Products of vector-valued forms. Directional derivative of a vector-valued function. Exterior derivative of a vector-valued form. Differential forms with values in a Lie algebra. Pullback of vector-valued forms. Forms with values in a vector bundle. Tensor fields on a manifold. The tensor criterion.
- Connections and curvature againConnection and curvature matrices under a change of frame. Bianchi identities. The first Bianchi identity in vector form. Symmetry properties of the curvature tensor. Covariant derivative of tensor fields. The second Bianchi identity in vector form. Ricci curvature. Scalar curvature. Defining a connection using connection matrices. Induced connection on a pullback bundle.
- Characteristic classesInvariant polynomials on gl(r;R). The Chern-Weil homomorphism. Characteristic forms are closed. Differential forms depending on a real parameter. Independence of characteristic classes of a connection. Functorial definition of a characteristic class.
- Pontrjagin classes. The Euler class and Chern classes.Vanishing of characteristic classes. Pontrjagin classes. The Whitney product formula. Orientation on a vector bundle. Characteristic classes of an oriented vector bundle. The Pfaffian of a skew-symmetric matrix. The Euler class. Generalized Gauss-Bonnet theorem. Hermitian metrics. Connections and curvature on a complex vector bundle. Chern classes.
- Some applications of characteristic classesThe generalized Gauss-Bonnet theorem. Characteristic numbers. The cobordism problem. The embedding problem. The Hirzebruch signature formula. The Riemann-Roch.
- Principal bundlesPrincipal bundles. The frame bundle of a vector bundle. Fundamental vector fields of a right action. Integral curves of a fundamental vector field. Vertical subbundle of the tangent bundle TP. Horizontal distributions on a principal bundle.
- Connections on a principal bundleConnections on a principal bundle. Vertical and horizontal components of a tangent vector. The horizontal distribution of an Ehresmann connection. Horizontal lift of a vector field to a principal bundle. Lie bracket of a fundamental vector field. Horizontal distributions on a frame bundle. Parallel translation in a vector bundle. Horizontal vectors on a frame bundle. Horizontal lift of a vector field to a frame bundle. Pullback of a connection on a frame bundle under a section.
- Curvature on a principal bundleCurvature form on a principal bundle. Properties of the curvature form. The associated bundle. The fiber of the associated bundle. Tensorial forms on a principal bundle. Covariant derivative. A formula for the covariant derivative of a tensorial form.
- Characteristic classes of principal bundlesInvariant polynomials on a Lie algebra. The Chern- Weil homomorphism

#### Interim Assessment

- Interim assessment (2 module)total grade = 0,3(grade for homework)+0,2(grade for midterm exam)+ 0,5(grade for final exam)

#### Bibliography

#### Recommended Core Bibliography

- Spivak, M. (1998). Calculus On Manifolds : A Modern Approach To Classical Theorems Of Advanced Calculus. New York: CRC Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=421137

#### Recommended Additional Bibliography

- Reed, M. (1972). Methods of Modern Mathematical Physics : Functional Analysis. Oxford: Academic Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=567963