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This book is an introduction to differential manifolds. It gives solid preliminaries for more advanced topics: Riemannian manifolds, differential topology, Lie theory. It presupposes little background: the reader is only expected to master basic differential calculus, and a little point-set topology. The book covers the main topics of differential geometry: manifolds, tangent space, vector fields, differential forms, Lie groups, and a few more sophisticated topics such as de Rham cohomology, degree theory and the Gauss-Bonnet theorem for surfaces.

Its ambition is to give solid foundations. In particular, the introduction of “abstract” notions such as manifolds or differential forms is motivated via questions and examples from mathematics or theoretical physics. More than 150 exercises, some of them easy and classical, some others more sophisticated, will help the beginner as well as the more expert reader. Solutions are provided for most of them.

The book should be of interest to various readers: undergraduate and graduate students for a first contact to differential manifolds, mathematicians from other fields and physicists who wish to acquire some feeling about this beautiful theory.

The original French text Introduction aux variétés différentielles has been a best-seller in its category in France for many years.

Jacques Lafontaine was successively assistant Professor at Paris Diderot University and Professor at the University of Montpellier, where he is presently emeritus. His main research interests are Riemannian and pseudo-Riemannian geometry, including some aspects of mathematical relativity. Besides his personal research articles, he was involved in several textbooks and research monographs.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Differential Calculus

In this chapter, we review and reinforce the basics of differential calculus in preparation for our subsequent study of manifolds.

Jacques Lafontaine

Chapter 2. Manifolds: The Basics

“The notion of a manifold is hard to define precisely.” This is the famous opening of Chapter III of

Leçons sur la Géométrie des espaces de Riemann

by Elie Cartan. It is followed by a stimulating heuristic discussion on the notion of manifold which can still be read with pleasure. For additional historic perspective we also mention Riemann’s inaugural lecture, translated with annotations for the modern reader in [Spivak 79].

Jacques Lafontaine

Chapter 3. From Local to Global

This chapter consists of variations on the following themes:

Jacques Lafontaine

Chapter 4. Lie Groups

The notion of group was singled out around 1830 by Évariste Galois in his work on algebraic equations. This initial work was with finite groups.

Forty years later, the work of Galois inspired the Norwegian mathematician Sophus Lie, who rather than studying invariance of algebraic equations was studying the invariance properties of ordinary and partial differential equations and put the need for other types of groups into focus. These were formerly called “finite and continuous groups”, which in today’s language conveys groups of finite topological dimension. In fact many of the examples discovered were smooth manifolds, with smooth group operations. Today we call such groups Lie groups.

Jacques Lafontaine

Chapter 5. Differential Forms

Does there exist a theory of integration – first for

p

-dimensional submanifolds of Euclidean space –, and more generally for manifolds? We can start with what we call line integrals, which is to say the circulation of a vector field

V

along a curve. This is classically defined as the integral

a

b

V

c

(

t

)

,

c

(

t

)

d

t

.

$$\displaystyle{\int _{a}^{b}\big\langle V _{ c(t)},c^{{\prime}}(t)\big\rangle \,dt.}$$

Here,

c

:

[

a

,

b

]

R

n

$$c:\, [a,b] \rightarrow \mathbf{R}^{n}$$

is a curve parametrization (in fact a piece of the parametrization as we restricted the parameter to the interval [

a

, 

b

]) and

,

$$\langle \;,\,\rangle$$

is an inner product on

R

n

$$\mathbf{R}^{n}$$

. Replacing the vectors

V

c

(

t

)

by

linear forms α

c

(

t

)

has the advantage of no longer requiring the inner product. We can then integrate curves on any manifold

X

, the “field of linear forms”

x

α

x

, for all

x

 ∈ 

X

, where

α

x

is a linear form on the tangent space

T

x

X

, by writing

c

α

=

a

b

α

c

(

t

)

c

(

t

)

d

t

.

$$\displaystyle{\int _{c}\alpha =\int _{ a}^{b}\alpha _{ c(t)}\big(c^{{\prime}}(t)\big)\,dt.}$$

Jacques Lafontaine

Chapter 6. Integration and Applications

If

f

is a function of a real variable with continuous derivative,

a

b

f

(

t

)

d

t

=

f

(

b

)

f

(

a

)

.

$$\displaystyle{\int _{a}^{b}f^{{\prime}}(t)\,dt = f(b) - f(a).}$$

Jacques Lafontaine

Chapter 7. Cohomology and Degree Theory

In the preceding chapters we saw several ways to show that two open subsets of

R

n

$$\mathbf{R}^{n}$$

, and more generally two manifolds, are not diffeomorphic.

Jacques Lafontaine

Chapter 8. The Euler-Poincaré Characteristic and the Gauss-Bonnet Theorem

The Gauss-Bonnet theorem is at the heart of the geometry of manifolds. It mixes topology (triangulations, cohomology spaces), differential geometry (index of singular points of vector fields) and Riemannian geometry. We do not have the space to illustrate all of these ideas in detail. To keep with the spirit of the book, the proofs we give will use differential geometry to the greatest extent possible. We nonetheless believe it would be interesting to sketch a purely Riemannian proof in this introduction. The price we pay is using certain notions that have not been introduced (geodesics, geodesic curvature), of which we give the idea.

Jacques Lafontaine

Backmatter

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