# Differentiable Manifolds Level 4 and postgraduate (15 credits)

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

Course webpage: `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.)

Description:

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 Rn, and include "multi-dimensional surfaces" such as the n-sphere Sn and n-torus Tn, the projective spaces RPn and CPn, 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

Information about the exam: Exam will consist of sections A and B. Section A is compulsory (you will have to answer ALL FOUR questions, 60 marks total). It will contain theoretical questions and examples on basics of manifolds (charts, atlases, changes of coordinates,...), tangent vectors, velocity, bases in tangent space, vector fields, commutators..., differential forms and exterior differential (axioms and calculation; closed forms, exact forms, and examples when a closed form may be not exact), basics of de Rham cohomology (definition, pullbacks, basic properties, examples for simple manifolds). In section B, you will have to answer TWO of the THREE questions (40 marks total). They are slightly more advanced compared to section A. Topics: constructions of manifolds such as direct products, open and closed submanifolds, specifying manifolds by equations (cases of non-degenerate equations and constant rank, application of the auxiliary linear system); derivations and derivations over an algebra homomorphism, Hadamard's lemma, vectors and vector fields as derivations; properties of commutator of vector fields; construction of integral over a manifold, in particular, independence of choices, Stokes theorem (statement in different variants and proofs) and relation with being closed/ exact form; calculation of de Rham cohomology using its properties such as homotopy invariance and Poincaré lemma (proofs of the latter two properties not required). Each question contains a problem so that you may show how you are able to apply your theoretical knowledge.