黎曼幾何

出版時(shí)間:2008-1  出版社:世界圖書出版公司  作者:加洛特  頁(yè)數(shù):322  
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內(nèi)容概要

本書是一部值得一讀的研究生教材,內(nèi)容主要涉及黎曼幾何基本定理的研究,如霍奇定理、rauch比較定理、lyusternik和fet定理調(diào)和映射的存在性等。另外,書中還有當(dāng)代數(shù)學(xué)研究領(lǐng)域中的最熱門論題,有些內(nèi)容則是首次出現(xiàn)在教科書中。該書適合數(shù)學(xué)和理論物理專業(yè)的研究生、教師和科研人員閱讀研究。

書籍目錄

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

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  •   這本書是很好的黎曼幾何教材,適合數(shù)學(xué)研究生閱讀。第3版增加了pseudo-Riemanniangeomety的內(nèi)容。
  •   影印的挺清晰,你值得擁有!此乃廣義相對(duì)論兩大神器之一,是上乘武功秘籍。
  •   書是很好
  •   很好的教材,感覺比起國(guó)內(nèi)的教材看著舒服。推薦
  •   經(jīng)典之作。
    原版的影印,質(zhì)量不錯(cuò)
  •   這本書可以作為黎曼幾何的書讀,比較好,作者都很經(jīng)典啊
  •   叢書名雖然是UniversiTextinMathematics(UTM,區(qū)別于Springer的GTM),但實(shí)際上主要讀者對(duì)象仍然是數(shù)學(xué)類和物理類的研究生。以前的這類書國(guó)內(nèi)常常會(huì)有高校組織相關(guān)專家翻譯出來(lái)以供教學(xué)使用,現(xiàn)在,研究生的英語(yǔ)水平提高了,可以直接閱讀原著了,所以很多引進(jìn)的GTM和UTM書籍都沒有在組織翻譯。當(dāng)然,閱讀是難免會(huì)遇到很多不認(rèn)識(shí)的專業(yè)名詞,一般的英漢辭典都不能夠查。這種情況下,就只有買專門的電子辭典或上網(wǎng)查詢了。如果能夠靜下心來(lái)把這本書讀完,你會(huì)發(fā)現(xiàn),不久你的數(shù)學(xué)水平大有長(zhǎng)進(jìn),英語(yǔ)水平也有很大提高。當(dāng)然,看這本書不能夠象看小說(shuō)一樣一目十行,而要字斟句酌,每一個(gè)證明都要弄明白,每一個(gè)概念都要搞清楚。否則,讀與不讀沒有什么差別。
  •   適合有初級(jí)基礎(chǔ)的人,最好看完微分幾何入門書后再看這本書,感覺還可以。
  •   上次買的書質(zhì)量比較次,這次的還行
  •   慢慢看,術(shù)語(yǔ)太多
 

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