Ce livre aborde les variétés riemanniennes, en explorant leurs invariants et le spectre associé. Il offre une analyse approfondie des propriétés géométriques et spectrales, essentielle pour les mathématiciens et les physiciens intéressés par la géométrie différentielle et ses applications.
Both classical and modern differential geometry have been vital areas of research throughout the 20th century, central to many advancements in mathematics and physics. This book aims to cultivate a modern geometric culture through a series of visually appealing unsolved or recently solved problems that necessitate the development of concepts and tools of varying abstraction. It begins with fundamental objects such as lines, planes, circles, spheres, polygons, polyhedra, curves, and surfaces, elucidating crucial ideas and abstract concepts essential for achieving results. These concepts build progressively, akin to Jacob's ladder, allowing for increased abstraction. The book illustrates the enduring spirit of geometry, emphasizing that even "elementary" geometry remains vibrant and integral to the work of contemporary mathematicians. It highlights the countless unexplored paths and concepts waiting to be discovered. Rich in visual content, the book invites readers to engage with it at random, offering enjoyment based on their own intuitions and interests. Authored by Marcel Berger, a prolific writer on geometry, this work is designed for students and teachers of mathematics who have a passion for the subject.
Riemannian geometry has today become a vast and important subject. This new book of Marcel Berger sets out to introduce readers to most of the living topics of the field and convey them quickly to the main results known to date. These results are stated without detailed proofs but the main ideas involved are described and motivated. This enables the reader to obtain a sweeping panoramic view of almost the entirety of the field. However, since a Riemannian manifold is, even initially, a subtle object, appealing to highly non-natural concepts, the first three chapters devote themselves to introducing the various concepts and tools of Riemannian geometry in the most natural and motivating way, following in particular Gauss and Riemann.
This book serves as a companion to Marcel Berger's Geometry I and II, featuring a variety of exercises and solutions aligned with each chapter of the original volumes. It starts with accessible material, includes hints for problems, and covers topics from affine spaces to conics, making it a useful resource for geometry studies.
This is the second of a two-volume textbook that provides a very readable and lively presentation of large parts of geometry in the classical sense. For each topic the author presents a theorem that is esthetically pleasing and easily stated, although the proof may be quite hard and concealed. Yet another strong trait of the book is that it provides a comprehensive and unified reference source for the field of geometry in the full breadth of its subfields and ramifications.
InhaltsverzeichnisVarietes C ? — Varietes Riemanniennes.Varietes C ?.Varietes Kahleriennes.Eclatements.Cohomologie Et Formes Harmoniques.Cohomologie Des Varietes Kahleriennes.Espaces Fibres Vectoriels.C ? Fibres En Droites.Surfaces De Riemann.Theoreme De Kodaira.Connexions.Classes De Chern.