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## Über dieses Buch

In this second edition, the main additions are a section devoted to surfaces with constant negative curvature, and an introduction to conformal geometry. Also, we present a -soft-proof of the Paul Levy-Gromov isoperimetric inequal­ ity, kindly communicated by G. Besson. Several people helped us to find bugs in the. first edition. They are not responsible for the persisting ones! Among them, we particularly thank Pierre Arnoux and Stefano Marchiafava. We are also indebted to Marc Troyanov for valuable comments and sugges­ tions. INTRODUCTION This book is an outgrowth of graduate lectures given by two of us in Paris. We assume that the reader has already heard a little about differential manifolds. At some very precise points, we also use the basic vocabulary of representation theory, or some elementary notions about homotopy. Now and then, some remarks and comments use more elaborate theories. Such passages are inserted between *. In most textbooks about Riemannian geometry, the starting point is the local theory of embedded surfaces. Here we begin directly with the so-called "abstract" manifolds. To illustrate our point of view, a series of examples is developed each time a new definition or theorem occurs. Thus, the reader will meet a detailed recurrent study of spheres, tori, real and complex projective spaces, and compact Lie groups equipped with bi-invariant metrics. Notice that all these examples, although very common, are not so easy to realize (except the first) as Riemannian submanifolds of Euclidean spaces.

## Inhaltsverzeichnis

### Chapter I. Differential Manifolds

Abstract
“The general notion of manifold is quite difficult to define precisely. A surface gives the idea of a two-dimensional manifold. If we take for instance a sphere, or a torus, we can decompose this surface into a finite number of parts such that each of them can be bijectively mapped into a simply-connected region of the Euclidean plane.” This is the beginning of the third chapter of “Leçons sur la Géométrie des espaces de Riemann” by Elie Cartan (1928), that we strongly recommand to those who can read French. In fact, Cartan explains very neatly that these parts are what we call “open sets”. He explains also that if the domains of definition of two such maps (which are now called charts) overlap, one of them is gotten from the other by composition with a smooth map of the Euclidean space. This is just the formal definition of a differential (or smooth) manifold that we give in paragraph A, and illustrate by the examples of the sphere and the torus, and also the projective spaces.
Sylvestre Gallot, Dominique Hulin, Jacques Lafontaine

### Chapter II. Riemannian Metrics

Abstract
The Pythagorus theorem just says that the squared length of an infinitesimal vector, say in R 3, whose components are dx, dy and dz, is dx 2 + dy 2 + dz 2. Thus, the length of a parameterized curve c(t) = (x(t), y(t), z(t)) is given by the integral
$$\int {ds = \int {{{\left( {{x^{{'2}}} + {y^{{'2}}} + {z^{{'2}}}} \right)}^{{1/2}}}} } dt$$
.
Sylvestre Gallot, Dominique Hulin, Jacques Lafontaine

### Chapter III. Curvature

Abstract
A parallel vector field in R 2 is just a constant field. Now, on a surface, there are generally no (even local) parallel vector fields. How much the parallel transport of a field along a small closed curve differ from the identity is measured in terms of the curvature of the surface, a function k: MR.
Sylvestre Gallot, Dominique Hulin, Jacques Lafontaine

### Chapter IV. Analysis on Manifolds and the Ricci Curvature

Abstract
Analysis on Riemannian manifolds stems from the following simple fact: the classical Laplace operator has an exact Riemannian analog. Indeed, the properties of the Laplacian on a bounded Euclidean domain and on a compact Riemannian manifold are very similar, and so are the techniques of proofs. We can say that the difficulties of the latter case, compared with the former, are essentially conceptual.
Sylvestre Gallot, Dominique Hulin, Jacques Lafontaine

### Chapter V. Riemannian Submanifolds

Abstract
In this chapter, we study the relations between the Riemannian Geometry of a submanifold and that of the ambiant space. It is well known that surfaces of the Euclidean space were the first examples of Riemannian manifolds to be studied. In fact, the first truly Riemannian geometry result is due to Gauss, and roughly says the following.
Sylvestre Gallot, Dominique Hulin, Jacques Lafontaine

### Backmatter

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