黎曼几何

出版时间：2008-1  出版社：世界图书出版公司  作者：加洛特  页数：322  
Tag标签：无  

内容概要

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。

书籍目录

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 unity2 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 ellipticity3 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　3.B First and second variation　　3.B.1 Technical preliminaries　　3.B.2 First variation formula　　3.B.3 Second variation formula　3.C Jacobi vector fields　　3.C.1 Basic topics about second derivatives　　3.C.2 Index form　　3.C.3 Jacobi fields and exponential map　　3.C.4 Applications　3.D Riemannian submersions and curvature　　3.D.1 Riemannian submersions and connections　　3.D.2 Jacobi fields of PnC　　3.D.3 O'Neill's formula　　3.D.4 Curvature and length of small circles.Application to Riemannian submersions　3.E The behavior of length and energy in the neighborhood of a geodesic　　3.E.1 Gauss lemma　　3.E.2 Conjugate points　　3.E.3 Some properties of the cut-locus　3.F Manifolds with constant sectional curvature　3.G Topology and curvature: two basic results　　3.G.1 Myers' theorem　　3.G.2 Cartan-Hadamard's theorem　3.H Curvature and volume　　3.H.1 Densities on a differentiable manifold　　3.H.2 Canonical measure of a Riemannian manifold　　3.H.3 Examples: spheres, hyperbolic spaces, complex projective spaces　　3.H.4 Small balls and scalar curvature　　3.H.5 Volume estimates　3.I Curvature and growth of the fundamental group　　3.I.1 Growth of finite type groups　　3.I.2 Growth of the fundamental group of compact manifolds with negative curvature　3.J Curvature and topology: some important results　　3.J.1 Integral formulas　　3.J.2 (Geo)metric methods　　3.J.3 Analytic methods　　3.J.4 Coarse point of view: compactness theorems　3.K Curvature tensors and representations of the orthogonal group　　3.K.1 Decomposition of the space of curvature tensors　　3.K.2 Conformally flat manifolds　　3.K.3 The Second Bianchi identity　3.L Hyperbolic geometry　　3.L.1 Introduction　　3.L.2 Angles and distances in the hyperbolic plane　　3.L.3 Polygons with "many" right angles　　3.L.4 Compact surfaces　　3.L.5 Hyperbolic trigonometry　　3.L.6 Prescribing constant negative curvature　　3.L.7 A few words about higher dimension　3.M Conformal geometry　　3.M.1 Introduction　　3.M.2 The MSbius group　　3.M.3 Conformal, elliptic and hyperbolic geometry4 Analysis on manifolds　4.A Manifolds with boundary　　4.A.1 Definition　　4.A.2 Stokes theorem and integration by parts　4.B Bishop inequality　　4.B.1 Some commutation formulas　　4.B.2 Laplacian of the distance function.　　4.B.3 Another proof of Bishop's inequality　　4.B.4 Heintze-Karcher inequality　4.C Differential forms and cohomology　　4.C.1 The de Rham complex　　4.C.2 Differential operators and their formal adjoints　　4.C.3 The Hodge-de Rham theorem　　4.C.4 A second visit to the Bochner method　4.D Basic spectral geometry　　4.D.1 The Laplace operator and the wave equation　　4.D.2 Statement of basic results on the spectrum　4.E Some examples of spectra　　4.E.1 Introduction　　4.E.2 The spectrum of flat tori　　4.E.3 Spectrum of (Sn,can)　4.F The minimax principle　4.G Eigenvalues estimates　　4.G.1 Introduction　　4.G.2 Bishop's inequality and coarse estimates　　4.0.3 Some consequences of Bishop's theorem　　4.G.4 Lower bounds for the first eigenvalue　4.H Paul Levy's isoperimetric inequality　　4.H.1 The statement　　4.H.2 The proof5 Riemannian submanifolds　5.A Curvature of submanifolds　　5.A.1 Second fundamental form　　5.A.2 Curvature of hypersurfaces　　5.A.3 Application to explicit computations of curvatures　5.B Curvature and convexity　5.C Minimal surfaces　　5.C.1 First results　　5.C.2 Surfaces with constant mean curvature　A Some extra problems　B Solutions of exercisesBibliographyIndexList of figures

用户评论 (总计10条)

•   这本书是很好的黎曼几何教材，适合数学研究生阅读。第3版增加了pseudo-Riemanniangeomety的内容。
•   影印的挺清晰，你值得拥有！此乃广义相对论两大神器之一，是上乘武功秘籍。
•   书是很好
•   很好的教材，感觉比起国内的教材看着舒服。推荐
•   经典之作。
  原版的影印，质量不错
•   这本书可以作为黎曼几何的书读,比较好,作者都很经典啊
•   丛书名虽然是UniversiTextinMathematics（UTM，区别于Springer的GTM），但实际上主要读者对象仍然是数学类和物理类的研究生。以前的这类书国内常常会有高校组织相关专家翻译出来以供教学使用，现在，研究生的英语水平提高了，可以直接阅读原著了，所以很多引进的GTM和UTM书籍都没有在组织翻译。当然，阅读是难免会遇到很多不认识的专业名词，一般的英汉辞典都不能够查。这种情况下，就只有买专门的电子辞典或上网查询了。如果能够静下心来把这本书读完，你会发现，不久你的数学水平大有长进，英语水平也有很大提高。当然，看这本书不能够象看小说一样一目十行，而要字斟句酌，每一个证明都要弄明白，每一个概念都要搞清楚。否则，读与不读没有什么差别。
•   适合有初级基础的人，最好看完微分几何入门书后再看这本书，感觉还可以。
•   上次买的书质量比较次,这次的还行
•   慢慢看，术语太多