Chavel - Riemannian Geometry - Modern Introduction 2e (Cambridge, 2006).pdf - Geometry - Kuya

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RIEMANNIAN GEOMETRY

A Modern Introduction

Second Edition

This book provides an introduction to Riemannian geometry, the geometry of

curved spaces, for use in a graduate course. Requiring only an understanding of

differentiable manifolds, the book covers the introductory ideas of Riemannian

geometry, followed by a selection of more specialized topics. Also featured

are Notes and Exercises for each chapter to develop and enrich the reader’s

appreciation of the subject. This second edition has a clearer treatment of many

topics from the ﬁrst edition, with new proofs of some theorems. Also a new

chapter on the Riemannian geometry of surfaces has been added.

The main themes here are the effect of curvature on the usual notions of

classical Euclidean geometry, and the new notions and ideas motivated by cur-

vature itself. Among the classical topics shown in a new setting is isoperimetric

inequalities – the interplay of volume of sets and the areas of their bound-

aries – in curved space. Completely new themes created by curvature include

the classical Rauch comparison theorem and its consequences in geometry and

topology, and the interaction of microscopic behavior of the geometry with the

macroscopic structure of the space.

Isaac Chavel is Professor of Mathematics at The City College of the City

University of New York. He received his Ph.D. in Mathematics from Yeshiva

University under the direction of Professor Harry E. Rauch. He has published in

international journals in the areas of differential geometry and partial differen-

tial equations, especially the Laplace and heat operators on Riemannian mani-

folds. His other books include Eigenvalues in Riemannian Geometry (1984) and

Isoperimetric Inequalities: Differential Geometric and Analytic Perspectives

(2001). He has been teaching at The City College of the City University of

New York since 1970, and he has been a member of the doctoral program of

the City University of New York since 1976. He is a member of the American

Mathematical Society.

CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS

Editorial Board:

B. Bollobas, W. Fulton, A. Katok, F. Kirwan, P. Sarnak, B. Simon, B. Totaro

Already published

17 W. Dicks & M. Dunwoody Groups acting on graphs

18 L.J. Corwin & F.P. Greenleaf Representations of nilpotent Lie groups and their applications

19 R. Fritsch & R. Piccinini Cellular structures in topology

20 H. Klingen Introductory lectures on Siegel modular forms

21 P. Koosis The logarithmic integral II

22 M.J. Collins Representations and characters of ﬁnite groups

24 H. Kunita Stochastic ﬂows and stochastic differential equations

25 P. Wojtaszczyk Banach spaces for analysis

26 J.E. Gilbert & M.A.M. Murray Clifford algebras and Dirac operators in harmonic analysis

27 A. Frohlich & M.J. Taylor Algebraic number theory

28 K. Goebel & W.A. Kirk Topics in metric ﬁxed point theory

29 J.F. Humphreys Reﬂection groups and Coxeter groups

30 D.J. Benson Representations and cohomology I

31 D.J. Benson Representations and cohomology II

32 C. Allday & V. Puppe Cohomological methods in transformation groups

33 C. Soul´eetal. Lectures on Arakelov geometry

34 A. Ambrosetti & G. Prodi A primer of nonlinear analysis

35 J. Palis & F. Takens Hyperbolicity, stability and chaos at homoclinic bifurcations

37 Y. Meyer Wavelets and operators I

38 C. Weibel An introduction to homological algebra

39 W. Bruns & J. Herzog Cohen–Macaulay rings

40 V. Snaith Explicit Brauer induction

41 G. Laumon Cohomology of Drinfeld modular varieties I

42 E.B. Davies Spectral theory and differential operators

43 J. Diestel, H. Jarchow, & A. Tonge Absolutely summing operators

44 P. Mattila Geometry of sets and measures in Euclidean spaces

45 R. Pinsky Positive harmonic functions and diffusion

46 G. Tenenbaum Introduction to analytic and probabilistic number theory

47 C. Peskine An algebraic introduction to complex projective geometry

48 Y. Meyer & R. Coifman Wavelets

49 R. Stanley Enumerative combinatorics I

50 I. Porteous Clifford algebras and the classical groups

51 M. Audin Spinning tops

52 V. Jurdjevic Geometric control theory

53 H. Volklein Groups as Galois groups

54 J. Le Potier Lectures on vector bundles

55 D. Bump Automorphic forms and representations

56 G. Laumon Cohomology of Drinfeld modular varieties II

57 D.M. Clark & B.A. Davey Natural dualities for the working algebraist

58 J. McCleary A user’s guide to spectral sequences II

59 P. Taylor Practical foundations of mathematics

60 M.P. Brodmann & R.Y. Sharp Local cohomology

61 J.D. Dixon et al. Analytic pro-p groups

62 R. Stanley Enumerative combinatorics II

63 R.M. Dudley Uniform central limit theorems

64 J. Jost & X. Li-Jost Calculus of variations

65 A.J. Berrick & M.E. Keating An introduction to rings and modules

66 S. Morosawa Holomorphic dynamics

67 A.J. Berrick & M.E. Keating Categories and modules with K-theory in view

68 K. Sato Levy processes and inﬁnitely divisible distributions

69 H. Hida Modular forms and Galois cohomology

70 R. Iorio & V. Iorio Fourier analysis and partial differential equations

71 R. Blei Analysis in integer and fractional dimensions

72 F. Borceaux & G. Janelidze Galois theories

73 B. Bollobas Random graphs

74 R.M. Dudley Real analysis and probability

75 T. Sheil-Small Complex polynomials

( continued on overleaf )

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