# MMath Mathematics with Financial Mathematics

Year of entry: 2023

## Course unit details:Differentiable Manifolds

Unit code MATH41061 15 Level 4 Semester 1 Department of Mathematics No

### Overview

Differentiable manifolds are among the most fundamental notions of modern mathematics. Roughly, they are geometrical objects that can be endowed with coordinates; using these coordinates one can apply differential and integral calculus, but the results are coordinate-independent.

Examples of manifolds start with open domains in Euclidean space <b>R</b><sup><i>n</i></sup>, and include "multidimensional surfaces" such as the <i>n</i>-sphere <i>S<sup>n</i></sup> and the <i>n</i>-torus i>T<sup>n</i></sup>, the projective spaces <b>RP</b><sup><i>n</i></sup> and <b>CP</b><sup><i>n</i></sup> and their generalizations, matrix groups such as the rotation group SO(<i>n</i>), etc. Differentiable manifolds naturally appear in various applications, e.g., as configuration spaces in mechanics. They are arguably the most general objects on which calculus can be developed. On the other hand, differentiable manifolds provide for calculus a powerful invariant geometric language, which is used in almost all areas of mathematics and its applications.

In this course we give an introduction to the theory of manifolds, including their definition and examples; vector fields and differential forms; integration on manifolds and de Rham cohomology.

### Pre/co-requisites

Unit title Unit code Requirement type Description
Topology MATH31052 Pre-Requisite Recommended
Metric Spaces MATH20122 Pre-Requisite Recommended
Calculus of Several Variables MATH20132 Pre-Requisite Recommended
MATH41061 pre-requisites

Students are not permitted to take MATH41062 and MATH61062 for credit in an undergraduate programme and then a postgraduate programme.

### Aims

The unit aims to introduce the fundamental notions of differentiable manifolds.

### Learning outcomes

On completion of this unit successful students will be able to:

• state the definition of a differentiable manifold and construct practically the manifold structure for various examples;
• state the definition of a tangent vector and find tangent spaces in examples;
• work practically with vector fields and differential forms; in particular, calculate commutator of vector fields and exterior differential of differential forms;
• state the definition of integral of a form, state the general Stokes theorem and calculate integrals of forms in particular examples with and without using the Stokes theorem;
• state the definition of de Rham cohomology, use its main properties to calculate the cohomology for particular examples, and apply it for distinguishing manifolds.

### Syllabus

1. Manifolds and smooth maps. Coordinates on familiar spaces. Charts and atlases. Definitions of manifolds and smooth maps. Products. Specifying manifolds by equations. More examples of manifolds.

2. Tangent vectors. Velocity of a curve. Tangent vectors. Tangent bundle. Differential of a map.

3. Topology of a manifold. Topology induced by manifold structure. Identification of tangent vectors with derivations. Bump functions and partitions of unity. Embedding manifolds in <b>R</b><sup><i>N</i></sup> .

4. Tensor algebra. Dual space, Covectors and tensors. Einstein notation. Behaviour under maps. Tensors at a point. Example: differential of a function as covector.

5. Vector fields. Tensor and vector fields. Examples. Vectors and vector fields as derivations. Flow of a vector field. Commutator.

6. Differential forms. Antisymmetric tensors. Exterior multiplication. Forms at a point. Bases and dimensions. Exterior differential: definition and properties.

7. Integration. Orientation. Integral over a compact oriented manifold. Independence of atlas and partition of unity. Integration over singular manifolds and chains. Stokes theorem in different variants.

8. De Rham cohomology. Definition of cohomology and examples of nonzero classes. Homotopy invariance. Poincar&eacute; Lemma. Examples of calculation.

### Assessment methods

Method Weight
Other 20%
Written exam 80%
• Coursework (take-home): weighting 20%
• End of semester examination: weighting 80%

### Feedback methods

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework also provides an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

No particular textbook is followed. Students are advised to keep their own lecture notes. There are many good sources available treating various aspects of differentiable manifolds on various levels and from different viewpoints. Below is a list of texts that may be useful. More can be found by searching library shelves.

• R. Abraham, J. E. Marsden, T. Ratiu. Manifolds, tensor analysis, and applications.
• 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. Morita. Geometry of differential forms.
• Michael Spivak. Calculus on manifolds.
• Frank W. Warner. Foundations of differentiable manifolds and Lie groups

### Study hours

Scheduled activity hours
Lectures 24
Tutorials 12
Independent study hours
Independent study 114

### Teaching staff

Staff member Role
Theodore Voronov Unit coordinator