黎曼几何（英文）（第3版）

本书是一部值得一读的研究生教材，内容主要涉及黎曼几何基本定理的研究，如霍奇定理、rauch比较定理、lyusternik和fet定理调和映射的存在性等。另外，书中还有当代数学研究领域中的最热门论题，有些内容则是首次出现在教科书中。该书适合数学和理论物理专业的研究生、教师和科研人员阅读研究。

本书是以作者在法国巴黎任教“黎曼几何和流行分析”研究生课程的讲义形成的。本书详细给出了曲率与拓扑学之间关系的经典结果，图文并茂，直观清晰。内容包括微分流行、黎曼度量、Levi-Civita连通、测地线和曲率，并特别强调他们的内蕴性质。
这是第三版，增加了一些有关测地流和Lorentzian几何的内容。

1 Differential manifolds
　1.A From submanifolds to abstract manifolds
　　1.A.1 Submanifolds of Euclidean spaces
　　1.A.2 Abstract manifolds
　　1.A.3 Smooth maps
　1.B The tangent bundle
　　1.B.1 Tangent space to a submanifold of Rn+k
　　1.B.2 The manifold of tangent vectors
　　1.B.3 Vector bundles
　　1.B.4 Tangent map
　1.C Vector fields
　　1.C.1 Definitions
　　1.C.2 Another definition for the tangent space
　　1.C.3 Integral curves and flow of a vector field
　　1.C.4 Image of a vector field by a diffeomorphism
　1.D Baby Lie groups
　　1.D.1 Definitions
　　1.D.2 Adjoint representation
　1.E Covering maps and fibrations
　　1.E.1 Covering maps and quotients by a discrete group
　　1.E.2 Submersions and fibrations
　　1.E.3 Homogeneous spaces
　1.F Tensors
　　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.G Differential forms
　　1.G.1 Definitions
　　1.G.2 Exterior derivative
　　1.G.3 Volume forms
　　1.G.4 Integration on an oriented manifold
　　1.G.5 Haar measure on a Lie group
　1.H Partitions of unity
2 Riemannian metrics
　2.A Existence theorems and first examples
　　2.A.1 Basic definitions
　　2.A.2 Submanifolds of Euclidean or Minkowski spaces
　　2.A.3 Riemannian submanifolds, Riemannian products
　　2.A.4 Riemannian covering maps, flat tori
　　2.A.5 Riemannian submersions, complex projective space
　　2.A.6 Homogeneous Riemannian spaces
　2.B Covariant derivative
　　2.B.1 Connections
　　2.B.2 Canonical connection of a Riemannian submanifold
　　2.B.3 Extension of the covariant derivative to tensors
　　2.B.4 Covariant derivative along a curve
　　2.B.5 Parallel transport
　　2.B.6 A natural metric on the tangent bundle
　2.C Geodesics
　　2.C.1 Definition, first examples
　　2.C.2 Local existence and uniqueness for geodesics,exponential map
　　2.C.3 Riemannian manifolds as metric spaces
　　2.C.4 An invitation to isosystolic inequalities
　　2.C.5 Complete Riemannian manifolds, Hopf-Rinow theorem.
　　2.C.6 Geodesics and submersions, geodesics of PnC:
　　2.C.7 Cut-locus
　　2.C.8 The geodesic flow
　2.D A glance at pseudo-Riemannian manifolds
　　2.D.1 What remains true?
　　2.D.2 Space, time and light-like curves
　　2.D.3 Lorentzian analogs of Euclidean spaces, spheres and hyperbolic spaces
　　2.D.4 (In)completeness
　　2.D.5 The Schwarzschild model
　　2.D.6 Hyperbolicity versus ellipticity
3 Curvature
　3.A The curvature tensor
　　3.A.1 Second covariant derivative
　　3.A.2 Algebraic properties of the curvature tensor
　　3.A.3 Computation of curvature: some examples
　　3.A.4 Ricci curvature, scalar curvature
……
4 Analysis on manifolds
5 Riemannian submanifolds
A Some extra problems
B Solutions of exercises
Bibliography
Index
List of figures