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Geometry VI

Riemannian Geometry

  • Textbook
  • © 2001

Overview

Authors:
  1. M. M. Postnikov

    1. MIRAN, Moscow, Russia

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  • A well-written introduction to the subject
  • Includes supplementary material: sn.pub/extras

Part of the book series: Encyclopaedia of Mathematical Sciences (EMS, volume 91)

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Table of contents (37 chapters)

  1. Front Matter

    Pages I-XVIII
    Download chapter PDF

  2. Affine Connections

    • M. M. Postnikov
    Pages 1-13

  3. Covariant Differentiation. Curvature

    • M. M. Postnikov
    Pages 14-28

  4. Affine Mappings. Submanifolds

    • M. M. Postnikov
    Pages 29-43

  5. Structural Equations. Local Symmetries

    • M. M. Postnikov
    Pages 44-54

  6. Symmetric Spaces

    • M. M. Postnikov
    Pages 55-66

  7. Connections on Lie Groups

    • M. M. Postnikov
    Pages 67-76

  8. Lie Functor

    • M. M. Postnikov
    Pages 77-86

  9. Affine Fields and Related Topics

    • M. M. Postnikov
    Pages 87-100

  10. Cartan Theorem

    • M. M. Postnikov
    Pages 101-113

  11. Palais and Kobayashi Theorems

    • M. M. Postnikov
    Pages 114-126

  12. Lagrangians in Riemannian Spaces

    • M. M. Postnikov
    Pages 127-140

  13. Metric Properties of Geodesics

    • M. M. Postnikov
    Pages 141-158

  14. Harmonic Functionals and Related Topics

    • M. M. Postnikov
    Pages 159-175

  15. Minimal Surfaces

    • M. M. Postnikov
    Pages 176-192

  16. Curvature in Riemannian Space

    • M. M. Postnikov
    Pages 193-206

  17. Gaussian Curvature

    • M. M. Postnikov
    Pages 207-222

  18. Some Special Tensors

    • M. M. Postnikov
    Pages 223-237

  19. Surfaces with Conformal Structure

    • M. M. Postnikov
    Pages 238-247

  20. Mappings and Submanifolds I

    • M. M. Postnikov
    Pages 248-261
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Keywords

About this book

The original Russian edition of this book is the fifth in my series "Lectures on Geometry. " Therefore, to make the presentation relatively independent and self-contained in the English translation, I have added supplementary chapters in a special addendum (Chaps. 3Q-36), in which the necessary facts from manifold theory and vector bundle theory are briefly summarized without proofs as a rule. In the original edition, the book is divided not into chapters but into lec­ tures. This is explained by its origin as classroom lectures that I gave. The principal distinction between chapters and lectures is that the material of each chapter should be complete to a certain extent and the length of chapters can differ, while, in contrast, all lectures should be approximately the same in length and the topic of any lecture can change suddenly in the middle. For the series "Encyclopedia of Mathematical Sciences," the origin of a book has no significance, and the name "chapter" is more usual. Therefore, the name of subdivisions was changed in the translation, although no structural surgery was performed. I have also added a brief bibliography, which was absent in the original edition. The first ten chapters are devoted to the geometry of affine connection spaces. In the first chapter, I present the main properties of geodesics in these spaces. Chapter 2 is devoted to the formalism of covariant derivatives, torsion tensor, and curvature tensor. The major part of Chap.

Reviews

From the reviews of the first edition:

"... The book is .. comprehensive and original enough to be of interest to any professional geometer, but I particularly recommend it to the advanced student, who will find a host of instructive examples, exercises and vistas that few comparable texts offer... "

H.Geiges, Nieuw Archief voor Wiskunde 2002, Vol. 5/3, Issue 4

"... I found the presentation insightful and stimulating. A useful paedagogical device of the text is to make much use of both the index and coordinate-free notations, encouraging flexibility (and pragmatism) in the reader. ... the book should be of use to a wide variaty of readers: the relative beginner, with perhaps an introductory course in differential geometry, will find his horizons greatly expanded in the material for which this prepares him; while the more experienced reader will surely find the more specialised sections informative."

Robert J. Low, Mathematical Reviews, Issue 2002g

"... Insgesamt liegt ein sehr empfehlenswertes Lehrbuch einerseits zur Riemannschen Geometrie und andererseits zur Theorie differenzierbarer Mannigfaltigkeiten vor, wegen der strukturierten Breite der Darstellung sehr gut geeignet sowohl zum Selbststudium für Studierende mathematischer Disziplinen als auch für Dozenten als Grundlage einschlägiger Lehrveranstaltungen."

P. Paukowitsch, Wien (IMN - Internationale Mathematische Nachrichten 190, S. 64-65, 2002)

"M.M. Postnikov has written a well-structured and readable book with a satisfying sense of completeness to it. The reviewer believes this book deserves a place next to the already existing literature on Riemannian geometry, principally as a basis for teaching a course on abstract Riemannian geometry (after an introduction to differentiable manifolds) but also as a reference work." (Eric Boeckx, Zentralblatt MATH, Vol. 993 (18), 2002)

Authors and Affiliations

  • MIRAN, Moscow, Russia

    M. M. Postnikov

Bibliographic Information

  • Book Title: Geometry VI

  • Book Subtitle: Riemannian Geometry

  • Authors: M. M. Postnikov

  • Series Title: Encyclopaedia of Mathematical Sciences

  • DOI: https://doi.org/10.1007/978-3-662-04433-9

  • Publisher: Springer Berlin, Heidelberg

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer-Verlag Berlin Heidelberg 2001

  • Hardcover ISBN: 978-3-540-41108-6Published: 13 March 2001

  • Softcover ISBN: 978-3-642-07434-9Published: 09 December 2010

  • eBook ISBN: 978-3-662-04433-9Published: 17 April 2013

  • Series ISSN: 0938-0396

  • Edition Number: 1

  • Number of Pages: XVIII, 504

  • Additional Information: Original Russian edition published by Faktorial, Moscow, 1998

  • Topics: Geometry, Differential Geometry

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