# 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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## Scans of my scrap notes 2005

You should treat them with all due disrespect: errors, omissions, etc are highly likely.

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§         Lecture 3

§         Lectures 4-5

§         Lectures 6-8

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## Exam (Spring semester, 2005).

#### 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.

## Bibliography

·  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!