Lecture notes
for the course in
Differential Geometry
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!)
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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.
§
Lecture 2
§
Lecture 3
§
Lectures 4-5
§
Lectures 6-8
§
Lectures
9-10
Exam
(Spring semester, 2005).
Due date: July 31, 2005. Good luck!
Bonus problem: Solve the
anagram
Elementary if it forged.
Lecture notes from the course first given in WIS
in 1992-1993 academic year and several times recycled since then.
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.
1. Introduction to
manifolds.
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.
1a.
Supplement to Lecture 1.
Matrix manifolds. Partitions of unity. Whitney (weak)
embedding theorem.
2. Tangent vectors. Tangent bundle.
Definition via
classes of first-order-equivalent curves. Tangent maps (differentials of
diffeomorphisms). Vector fields.
Action of diffeomorphisms on vector
fields.
3. Algebra of vector fields.
Lie derivatives.
Vector fields as differential
operators. Differential identities. Commutator as a
differential operator.
4. Algebraic language in
Geometry.
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.
5. Algebraic language in
Geometry (continued).
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).
6. Differential forms on
manifolds.
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.
8. Calculus versus topology.
Lie derivative of a differential
form. Homotopy formula. De Rham
cohomology of a smooth manifold. Examples.