Riemannian geometryRiemannian几何
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作者： Sylvestre Gallot等著
出 版 社： 北京燕山出版社
出版时间： 2004-12-1字数：版次： 1页数： 322印刷时间： 2004/12/01开本： 16开印次： 1纸张： 胶版纸I S B N ： 9783540204930包装： 平装内容简介
This book，based on a graduate course on Riemannian geometry and analysis on manifolds，held in Paris，covers the topics of differential manifolds，Riemannian metrics，connections，geodesics and curvature，with special emphasis on the intrinsic features of the subject。Classical results on the relations between curvature and topology are treated in detail。The book is quite self-contained，assuming of the reader only differential calculus in Euclidean space。It contains numerous exercises with full solutions and a series of detailed examples which are picked up repeatedly to illustrate each new definition or property introduced。
For this third edition，some topics about the geodesic flow and Lorentzian geometry have been added and worked out in the same spirit。
目录
1Differential manifolds
1.AFrom submanifolds to abstract manifolds
1.A.1Submanifolds of Euclidean spaces
1.A.2Abstract manifolds
1.A.3Smooth maps
1.B The tangent bundle
1.B.1Tangent space to a submanifold of Rn+k
1.B.2The manifold of tangent vectors
1.B.3Vector bundles
1.B.4Tangent map
1.CVector fields
1.C.1Definitions
1.C.2Another definition for the tangent space
1.C.3Integral curves and flow of a vector field
1.C.4Image of a vector field by a diffeomorphism
1.DBaby Lie groups
1.D.1Definitions
1.D.2Adjoint representation
1.ECovering maps and fibrations
1.E.1Covering maps and quotients by a discrete group
1.E.2Submersions and fibrations
1.E.3Homogeneous spaces
1.FTensors
1.F.1 Tensor product（a digest）
1.F.2 Tensor bundles
1.F.3 Operations on tensors
1.F.4 Lie derivatives
1.F.5 Local operators，differential operators
1.F.6 A characterization for tensors
1.GDifferential forms
1.G.1Definitions
1.G.2Exterior derivative
1.G.3Volume forms
1.G.4Integration on an oriented manifold
1.G.5Haar measure on a Lie group
1.HPartitions of unity
2Riemannian metrics
2.AExistence theorems and first examples
2.A.1Basic definitions
2.A.2Submanifolds of Euclidean or Minkowski spaces
2.A.3Riemannian submanifolds，Riemannian products
2.A.4Riemannian covering maps，fiat tori
2.A.5Riemannian submersions，complex projective space
2.A.6Homogeneous Riemannian spaces
2.BCovariant derivative
2.B.1Connections
2.B.2Canonical connection of a Riemannian submanifold
2.B.3Extension of the covariant derivative to tensors
2.B.4Covariant derivative along a curve
2.B.5Parallel transport
2.B.6A natural metric on the tangent bundle
2.CGeodesics
2.C.1Definition，first examples
2.C.2Local existence and uniquenessforgeodesics,exponential map
2.C.3Riemannian manifolds as metric spaces
2.C.4An invitation to isosystolic inequalities
2.C.5Complete Riemannian manifolds，Hopf-Rinow theorem
2.C.6Geodesics and submersions，geodesics of PnC:
2.C.7Cut-locus
2.C.8The geodesic flow
2.D A glance at pseudo-Riemannian manifolds
2.D.1What remains true?
2.D.2Space，time and light-like curves
2.D.3Lorentzian analogs of Euclidean spaces，spheres and
hyperbolic spaces
2.D.4 （In）completeness
2.D.5The Schwarzschild model
2.D.6Hyperbolicity versus ellipticity
3Curvature
3.A The curvature tensor
3.A.1 Second covariant derivative
3.A.2Algebraic properties of the curvature tensor
3.A.3Computation of curvature: some examples
3.A.4Ricci curvature，scalar curvature
……
4Analysis on manifolds
5 Riemannian submanifolds
A Some extra problems
B Solutions of exercises
Bibliography
Index
List of figrues