for the course in
Guided reading course for winter 2005/6*
The textbook: F. Warner, Foundations of Differentiable Manifolds and Lie Groups, Chapters 1, 2 and 4.
Take-home exam at the end of each semester (about 10-15 problems for four weeks of quiet thinking).
If you need additional reading, consider W. M. Boothby, Introduction to Differentiable Manifolds and Riemannian Geometry (Chapters I-VI)
*Books are in DejaVu format (download the plugin if you didn't do that yet!)
Besides, here are some fossils...
Scans of my scrap notes 2005
You should treat them with all due disrespect: errors, omissions, etc are highly likely.
Exam (Spring semester, 2005).
Bonus problem: Solve the anagram
Elementary if it forged.
Lecture notes from the course first given in
in 1992-1993 academic year and several times recycled since then. WIS
Mostly they constitute a collection of definitions, formulations of most important theorems and related problems for self-control.
Since that time, in 1996, I changed the order of exposition. Therefore the logical structure is not the same. Anyhow, I hope that these notes can still be useful for self-control. The general rule is always the same: if you do understand the problem, try to solve it. If you don't - disregard it.
Topolgical spaces, exotic topologies. Constructions (Cartesian product, quotient space, metric compatible with topology etc). Differentiable mappings of the Euclidean n-space. Diffeomorphisms. Definition of a smooth manifold.
Matrix manifolds. Partitions of unity. Whitney (weak) embedding theorem.
Definition via classes of first-order-equivalent curves. Tangent maps (differentials of diffeomorphisms). Vector fields. Action of diffeomorphisms on vector fields.
Vector fields as differential operators. Differential identities. Commutator as a differential operator.
Algebra of smooth functions as the Principal Example of An Algebra. Points as maximal ideals, diffeomorphisms as homomorphisms. Derivations. Exponential of a derivation. Algebraic nonsense versus common sense.
Maximal ideals. Local rings. Tangent and cotangent bundles a la maniere algebrique. Local algebra of a map, a function (preparations for introducing the notion of algebraic multiplicity).
Multilinear antisymmetric functionals on a linear n-space. Pullback of a multilinear form by a linear map. Wedge product. Relation to vector operations in R3
7. Exterior differential and integration of differential forms on manifolds.
Exterior derivative as the principal part of the integral over the boundary of an infinitesimal cell. Properties (linearity, d2=0). Stokes theorem.
Lie derivative of a differential form. Homotopy formula. De Rham cohomology of a smooth manifold. Examples.
Exams (version 1996)
· B. Dubrovin, S. Novikov, A. Fomenko, Modern Geometry, vol. 1.
· J. Milnor, Morse Theory.
For those who can read in Russian, here are the scanned translations in DejaVu format (download the plugin if you didn't do that yet!) of the two latter (and some other) books. (I am really sorry that such a treasure chest is not available in other languages...) Please print with discretion, save the Finnish forests and the Faculty expenses on toner cartridges!