**Lecturer: Dr Theodore Voronov**

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

**Classes in autumn semester 2018-2019:** (weeks 1-5 and 7-12)

**Mondays 9:00-9:50 (lecture 1): Alan Turing G.114**

**Mondays 15:00-15:50 (lecture 2): Alan Turing G.108**

**Fridays 10:00-10:50 (example class, starting week 2): Alan Turing G.209**
**Course webpage:** `http://personalpages.manchester.ac.uk/staff/theodore.voronov/manifolds.html`

(Page last modified: 23 August (5 September) 2018. Pages updated often, please refresh the browser to get the most recent version.)

**Prerequisites:** MATH20132 Calculus of Several Variables; MATH20222 Introduction to Geometry (*recommended*). (Also, MATH31052 Topology may be *beneficial* but is not required.)

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 will be posted HERE at the end of week 5 and will be due on Tuesday 13 November 2018 at 15:00.

**Syllabus and online materials: **

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

Differentiable manifolds are among the most fundamental notions of modern mathematics. Roughly, they are geometrical objects that can be endowed with coordinates; by using coordinates one can apply on them differential and integral calculus, but the results are coordinate-independent.
Examples of manifolds start with open domains in Euclidean space R^{n} and "multi-dimensional surfaces" such as the *n*-sphere S^{n} and *n*-torus T^{n}. In principle, every manifold of dimension *n* can be visualized as an "*n*-dimensional surface" in a Euclidean space R^{N} for some large *N*. Besides spheres and tori, classical examples include the projective spaces RP^{n} and CP^{n}, matrix groups such as the rotation group SO(n), the so-called Stiefel and Grassmann manifolds, and many others. Differentiable manifolds naturally arise in various applications, e.g., as configuration spaces of physical systems or as space-time. They are arguably the most general objects on which calculus can be developed and they provide for it 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 browsing 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: 23 August (5 September) 2018.
*Theodore Voronov.
*