**Lecturer: Dr Theodore Voronov**

Room 2.109 (Alan Turing Bldg). Email: `theodore.voronov@manchester.ac.uk`

**Classes in autumn semester 2017-2018:** (weeks 1-5 and 7-12)

**Tuesdays 3:00-3:50 pm (lecture): Williamson G.33
Thursdays 1:00-1:50 pm (lecture) Alan Turing G.108; 2:00-2:50 pm (example class, starts week 2): Alan Turing G.107**

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

(Page last modified: Thursday 1 (14) December 2017. Each time refresh the browser to get updated pages.)

**Prerequisites:** MATH20132 Calculus of Several Variables; MATH20222 Introduction to Geometry (*optional*). Taking in parallel MATH31051 Introduction to Topology may be beneficial but is optional.

See more details concerning **course description**, **prerequisites** and **textbooks** below.

**Assessment:**
Final mark = 20% coursework + 80% exam.

Coursework mode: take home work distributed in week 5; deadline beginning of week 8.

Exam mode: 3 hour exam at the end of semester.

**Coursework** (download it as a PDF file)
is due on Tuesday afternoon, 14 November 2017. (Please bring coursework to the lecture on Tuesday.)

**Syllabus and online materials: **

**Differentiable manifolds: definitions and examples.***Problems*.*Solutions*.-
**Differentiable maps.***Problems*.*Solutions*. **Constructions of manifolds.***Problems*.*Solutions*.**Tangent vectors and related objects.***Problems*.*Solutions*.**Manifolds as topological spaces.***Problems*.*Solutions*.**Vector fields. Tangent vectors as derivations.***Problems*.*Solutions*.**Differential forms: algebraic theory.***Problems*.*Solutions*.**Exterior differential.***Problems*.*Solutions*.**Integration of forms.***Problems*.*Solutions*.**De Rham cohomology.***Problems*.*Solutions*.

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 R^{n}, and include "multi-dimensional surfaces" such as the n-sphere S^{n} and n-torus T^{n}, the projective spaces RP^{n} and CP^{n}, matrix groups such as the rotation group SO(n), etc. Differentiable manifolds naturally arise in various applications, e.g., as configuration spaces of physical systems. 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.

** Details of prerequisites:**
Standard calculus and linear algebra; familiarity with the statement of the implicit function theorem (students may consult any good textbook in multivariate calculus); some familiarity with algebraic concepts such as groups and rings is desirable, but no knowledge of group theory or ring theory is assumed; some familiarity with basic topological notions (topological spaces, continuous maps, Hausdorff property, connectedness and compactness) may be beneficial, but is not assumed.

__ Textbooks:__
No particular textbook is followed. Students are advised to keep their own lecture notes and study my notes posted on the web. Solving problems is essential for understanding. 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
- S. Sternberg. Lectures on differential geometry

Last modified: Thursday 1 (14) December 2017.
*Theodore Voronov.
*