# Differentiable Manifolds

MATH31062: Level 3 (10 credits), MATH41062/ MATH61062: Level 4 and postgraduate (15 credits). Semester 2

Lecturer: Dr Theodore Voronov
Room 2.109 (Alan Turing Bldg). Email: theodore.voronov@manchester.ac.uk
Classes:
Tuesdays 2pm: Coupland 3, Theatre B,
Fridays 9am and 10am: Beyer, Beyer Theatre

Note: There will be revision classes in Week 12.

Enhanced version (15 credits): includes extra theoretical material and extra problems. Exam and coursework contain extra questions; exam is longer (3 hours).

Prerequisites: MATH20132 Calculus of Several Variables; MATH20222 Introduction to Geometry (optional); MATH31051 Introduction to Topology (optional)
Future topics requiring this course unit: differentiable manifolds are used in almost all areas of mathematics and its applications, including physics and engineering.

Details of prerequisites: standard calculus and linear algebra; familiarity with the statements of the implicit function theorem the existence and uniqueness theorem for ODEs (students may consult any good textbooks in multivariate calculus and differential equations); some familiarity with algebraic concepts such as groups and rings is desirable, but no knowledge of group theory or ring theory is assumed (students may refresh basic definitions using, e.g., J. Fraleigh, A First Course in Abstract Algebra); some familiarity with basic topological notions (topological spaces, continuous maps, Hausdorff property, connectedness and compactness) may be beneficial, but is not assumed (a good source is M. A. Armstrong, Basic Topology).

Last modified: Monday 8 (21) May 2012. (Refresh the browser to get the updated page.)

Brief 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, and their generalizations, matrix groups such as the rotation group SO(n), 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.

Textbooks:

No particular textbook is followed. Students are advised to keep their own lecture notes and use my notes posted on the web. 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

Assessment:

Final mark = 20% coursework + 80% exam.
Coursework mode: take home work distributed in week 6; deadline end of week 8.
Exam mode: 2 hour exam (10 credits) or 3 hour exam (15 credits) at the end of semester.

### Syllabus and online materials:

1. Differentiable manifolds and smooth maps
Lecture notes Problems. Solutions.
2. Constructions of manifolds
Lecture notes. Problems. Solutions.
3. Manifolds as topological spaces
Lecture notes. Problems. Solutions.
4. Tangent vectors
Lecture notes. Problems. Solutions.
5. Vector fields
Lecture notes. Problems. Solutions.
6. Differential forms
Lecture notes. Problems. Solutions.
7. Integration
Lecture notes. Problems. Solutions.
8. De Rham cohomology
Lecture notes. Problems. Solutions.

#### Coursework:

You can download the coursework here as a PDF file. Here you can see coursework solutions.

#### Exam:

##### Exam structure
The exam paper will consist of 4 questions (for 10 credit version) or 5 questions (for 15 credit version). You will have to answer any 3 questions out of Questions 1 to 4 (everybody). Those taking 15 credit version will also have to answer Question 5 (compulsory). Therefore, those taking 10 credit version will altogether answer 3 questions and those taking 15 credit version, 4 questions.
###### Questions 1 to 4
Each of them will consist of parts (a), (b), and (c):
(a) A definition or a group of related definitions and a question concerning a simple statement or example directly related with the definition.
(b) An important statement from the course (a theorem, or a lemma, or a part of a theorem). Possibly requiring some definition(s). You will be asked either to give the full statement and prove it; or quote a general statement and then prove some part of it; or you will be given a statement and you will have to give a proof; and/or you may be asked to deduce something from a general statement.
(c) A problem where you will have to calculate something or to show something about a concrete example.
###### Compulsory question 5 (15 credit version only)
(a) A definition and an important statement. You will have to quote some statements and give a proof.
(b) and (c) An advanced problem, subdivided into two parts.
##### Main exam topics
Questions 1 and 2. Charts, atlases, smoothness. Smooth manifolds and smooth maps. Diffeomorphism. Algebra of smooth functions. Specifying manifolds by equations. Submanifolds. Products. Tangent vectors and tangent spaces. Velocity of a parameterized curve. The natural basis of tangent vectors associated with a coordinate system. Tangent bundle. Partitions of unity.
Questions 3 and 4. Derivations of an algebra and derivations over an algebra homomorphism. Vectors and vector fields as derivations. Commutator of vector fields. Exterior differential: axiomatic definition and properties. Integration of forms and Stokes theorem. Closed and exact forms. De Rham cohomology: definition and examples. Pull-back and homotopy invariance of de Rham cohomology. Application to distinguishing manifolds.
Question 5. Embedding manifolds into RN. Existence of an embedding. Corollary from Sard's Lemma. Reducing the dimension of the ambient space (Whitney's Theorem). See § 4.3 of the online notes. Nota bene: This question will also include a more advanced problem concerning differential forms, Stokes theorem and de Rham cohomology.
##### Previous year paper with solutions:
`http://www.maths.manchester.ac.uk/~tv/manifolds.html`