具体描述
《黎曼几何概论》中着手本领域比较熟悉的话题，并且尽快过渡到最新科研成果。这些结果并没有给出详细的证明，但一些的重要的结果仍然描述的十分详细生动，使得读者对该领域有详细深刻的理解。然而，黎曼流形作为一个很小的科目，概念的建立需要一个过程，前三章侧重于通过常规的方法引入各种概念和黎曼几何的工具，紧接着详细讲述Gauss和Riemann。
黎曼几何概论 图书简介 本书旨在为读者提供一个严谨而全面的黎曼几何基础介绍。它是一部面向数学专业本科高年级学生、研究生以及对微分几何有浓厚兴趣的研究人员的教材或参考书。全书结构清晰，逻辑严密，力求在保持数学严谨性的同时，兼顾概念的直观性和可理解性。 核心内容概述： 全书从基础概念出发，逐步深入到黎曼几何的核心理论，涵盖了微分流形、张量分析、联络、曲率、测地线以及黎曼度量在几何结构中所扮演的关键角色。 第一部分：微分流形与张量分析基础 本部分为后续所有几何概念的构建打下坚实的分析和拓扑基础。 1. 流形的基础概念： 拓扑空间回顾与必要准备： 首先回顾紧凑性、连通性以及商拓扑等在流形构造中起关键作用的拓扑概念。 局部欧几里得空间与拓扑结构： 严格定义了 $n$ 维拓扑流形，强调其局部性质类似于 $mathbb{R}^n$。讨论了流形上的开集、坐标邻域、图册（Atlas）以及光滑性（可微性）的定义，特别是如何通过坐标变换确保不同图册之间的过渡映射是光滑的。 嵌入与淹没： 探讨了流形如何嵌入到更高维的欧几里得空间中，引入了嵌入定理的重要性。 2. 张量场与向量场： 切空间： 详细阐述了流形上每一点的切空间 $T_pM$ 的构造，将其理解为导子（derivation）的向量空间。 向量场与张量场： 定义光滑向量场及其在流形上的活动方式。引入张量场的概念，特别是张量积、对称张量与反对称张量。讨论了张量场在局部坐标系下的分量变换法则，这是区分张量与普通向量/函数集合的关键。 微分形式（Duals）： 引入 $p$ 阶协向量场（或微分 $p$ 形式），它们是切空间的对偶空间上的张量。详细讨论了楔积（Exterior Product）和外微分算子 $d$。 3. 微分算子与积分理论： 李导数： 阐述了李导数 $mathcal{L}_X$ 如何衡量一个向量场 $X$ 对一个几何对象（如函数、向量场或微分形式）的“拖拽”效应。 积分流形与流： 分析向量场产生的局部流（Flow）的性质。 Stokes 定理的推广： 在流形上讨论了广义 Stokes 定理，它将外微分与积分紧密联系起来，是微分形式理论的核心成果之一。 第二部分：联络与黎曼度量 本部分引入核心的几何结构——黎曼度量，并建立起“平行移动”的概念。 4. 黎曼流形与度量张量： 黎曼度量： 定义黎曼流形 $(M, g)$，其中 $g$ 是定义在每一点切空间上的一个光滑的、正定的、对称的二次型（内积）。讨论度量张量 $g$ 的局部坐标表示。 诱导长度与角度： 利用度量张量，可以精确计算流形上曲线的长度、向量之间的夹角，以及子流形的体积元。 5. 联络（Connection）的引入： 平行移动的必要性： 解释为什么在弯曲空间中，仅仅拥有切空间是不够的，需要一个机制来比较不同点的向量。 仿射联络： 引入仿射联络 $
abla$，它提供了一种“平行移动”向量的方法。讨论联络的必要性质：无挠性（Torsion-free）和度量兼容性（Metric compatibility）。 Levi-Civita 联络： 证明了对于任何黎曼流形，恰好存在一个唯一的联络，它既无挠又与度量 $g$ 兼容。本书将重点研究这个联络。 6. 协变导数与曲率： 协变导数： 利用 Levi-Civita 联络，定义协变导数 $
abla_X Y$，它是向量场 $Y$ 沿着向量场 $X$ 方向的变化率，是衡量流形弯曲的局部算子。 测地线方程： 将欧几里得空间中直线（导数为零）的概念推广到黎曼流形上，定义测地线为平行移动自身的曲线，并推导出其微分方程。 黎曼曲率张量： 明确定义黎曼曲率张量 $R(X, Y)Z$。该张量完全由度量 $g$ 及其一阶和二阶导数（即联络系数）决定。曲率张量是衡量流形弯曲程度的终极代数工具。 第三部分：几何的度量与应用 本部分利用曲率张量来分析流形的全局几何性质。 7. 曲率的度量： 截面曲率： 解释截面曲率（Sectional Curvature）的概念，它是通过曲率张量在任意二维切平面上导出的一个标量值，提供了对局部几何形状的直观理解。 里奇曲率与标量曲率： 定义里奇（Ricci）曲率张量（Ricci Tensor）作为黎曼曲率张量在特定方向上的缩并，并进一步定义标量曲率（Scalar Curvature）。讨论这些张量在爱因斯坦方程等物理应用中的重要性。 8. 测地线几何与全局性质： 测地线的存在性与唯一性： 证明了任意一点的任意速度向量都可以唯一地沿着一条测地线局部延伸，这是黎曼几何中的指数映射（Exponential Map）的基础。 变分原理： 将测地线解释为两点之间“能量”最小的路径，引入能量泛函，并利用泛函微分找到测地线方程。 卡坦-阿达马定理： 讨论了在恒定截面曲率空间（如球面和双曲空间）中，由于测地线行为的特定性所导致的全局拓扑限制。 9. 重要的黎曼流形： 常曲率空间： 深入分析 2 维和 3 维常截面曲率空间（如欧几里得空间、球面 $S^n$ 和双曲空间 $mathbb{H}^n$）的几何性质，以及它们在物理学和几何学中的地位。 全书特点： 本书的叙事风格力求从直觉出发，通过坐标表示辅助理解，但最终将所有结论建立在坐标无关的张量语言之上。每章后附有大量的习题，旨在巩固理论并引导读者进行计算实践。它不仅仅是一本关于如何计算曲率的指南，更是一部关于如何在弯曲空间中定义“直线”、“长度”和“角度”的精确数学手册。
作者简介
目录信息
《黎曼几何概论(英文影印版)》
1 euclidean geometry
1.1 preliminaries
1.2 distance geometry
1.2.1 a basic formula
1.2.2 the length of a path
1.2.3 the first variation formula and application to billiards
1.3 plane curves
1.3.1 length
1.3.2 curvature
1.4 global theory of closed plane curves
1.4.1 "obvious" truths about curves which are hard to prove
1.4.2 the four vertex theorem
1.4.3 convexity with respect to arc length
1.4.4 umlaufsatz with corners
1.4.5 heat shrinking of plane curves
1.4.6 arnol'd's revolution in plane curve theory
1.5 the isoperimetric inequality for curves
1.6 the geometry of surfaces before and after gaul}
1.6.1 inner geometry: a first attempt
.1.6.2 looking for shortest curves: geodesics
1.6.3 the second fundamental form and principal curvatures
1.6.4 the meaning of the sign of k
1.6.5 global surface geometry
1.6.6 minimal surfaces
1.6.7 the hartman-nirenberg theorem for inner flat surfaces
1.6.8 the isoperimetric inequality in e3 a la gromov
1.6.8.1 notes
1.7 generic surfaces
1.8 heat and wave analysis in e2
1.8.1 planar physics
1.8.1.1 bibliographical note
1.8.2 why the eigenvalue problem?
1.8.3 minimax
1.8.4 shape of a drum
1.8.4.1 a few direct problems
1.8.4.2 the faber-krahn inequality
1.8.4.3 inverse problems
1.8.5 heat
1.8.5.1 eigenfunctions
1.8.6 relations between the two spectra
1.9 heat and waves in e3, ed and on the sphere
1.9.1 euclidean spaces
1.9.2 spheres
1.9.3 billiards in higher dimensions
1.9.4 the wave equation versus the heat equation
2 transition
3 surfaces from gaufi to today
3.1 gau6
3.1.1 theorema egregium
3.1.1.1 the first proof of ganβ's theorema egregium; the concept of ds2
3.1.1.2 second proof of the theorema egregium.
3.1.2 the gang-bonnet formula and the rodrigues-gauiβ map
3.1.3 parallel transport
3.1.4 inner geometry
3.2 alexandrov's theorems
3.2.1 angle corrections of legendre and gauβ in geodesy
3.3 cut loci
3.4 global surface theory
3.4.1 bending surfaces
3.4.1.1 bending polyhedra
3.4.1.2 bending and wrinkling with little smoothness
3.4.2 mean curvature rigidity of the sphere
3.4.3 negatively curved surfaces
3.4.4 the willmore conjecture
3.4.5 the global gautβ-bonnet theorem for surfaces
3.4.6 the hopf index formula
4 riemann's blueprints
4.1 smooth manifolds
4.1.1 introduction
4.1.2 the need for abstract manifolds
4.1.3 examples
4.1.3.1 submanifolds
4.1.3.2 products
4.1.3.3 lie groups
4.1.3.4 homogeneous spaces
4.1.3.5 orassmannians over various algebras
4.1.3.6 gluing
4.1.4 the classification of manifolds
4.1.4.1 surfaces
4.1.4.2 higher dimensions
4.1.4.3 embedding manifolds in euclidean space
4.2 calculus on manifolds
4.2.1 tangent spaces and the tangent bundle
4.2.2 differential forms and exterior calculus
4.3 examples of riemann's definition
4.3.1 riemann's definition
4.3.2 hyperbolic geometry
4.3.3 products, coverings and quotients
4.3.3.1 products
4.3.3.2 coverings
4.3.4 homogeneous spaces
4.3.5 symmetric spaces
4.3.5.1 classification
4.3.5.2 rank
4.3.6 riemannian submersions
4.3.7 gluing and surgery
4.3.7.1 gluing of hyperbolic surfaces
4.3.7.2 higher dimensional gluing
4.3.8 classical mechanics
4.4 the riemann curvature tensor
4.4.1 discovery and definition
4.4.2 the sectional curvature
4.4.3 standard examples
4.4.3.1 constant sectional curvature
4.4.3.2 projective spaces kpn
4.4.3.3 products
4.4.3.4 homogeneous spaces
4.4.3.5 hypersurfaces in euclidean space
4.5 a naive question: does the curvature determine the metric?
4.5.1 surfaces
4.5.2 any dimension
4.6 abstract riemannian manifolds
4.6.1 isometrically embedding surfaces in e3
4.6.2 local isometric embedding of surfaces in ms
4.6.3 isometric embedding in higher dimensions
5 a one page panorama
6 metric geometry and curvature
6.1 first metric properties
6.1.1 local properties
6.1.2 hopf-rinow and de rham theorems
6.1.2.1 products
6.1.3 convexity and small balls
6.1.4 totally geodesic submanifolds
6.1.5 center of mass
6.1.6 examples of geodesics
6.1.7 transition
6.2 first technical tools
6.3 second technical tools
6.3.1 exponential map
6.3.1.1 rank
6.3.2 space forms
6.3.3 nonpositive curvature
6.4 triangle comparison theorems
6.4.1 bounded sectional curvature
6.4.2 ricci lower bound
6.4.3 philosophy behind these bounds
6.5 injectivity, convexity radius and cut locus
6.5.1 definition of cut points and injectivity radius
6.5.2 klingenberg and cheeger theorems
6.5.3 convexity radius
6.5.4 cut locus
6.5.5 blaschke manifolds
6.6 geometric hierarchy
6.6.1 the geometric hierarchy
6.6.1.1 space forms
6.6.1.2 rank 1 symmetric spaces
6.6.1.3 measure isotropy
6.6.1.4 symmetric spaces
6.6.1.5 homogeneous spaces
6.6.2 constant sectional curvature
6.6.2.1 negatively curved space forms in three and higher dimensions
6.6.2.2 mostow rigidity
6.6.2.3 classification of arithmetic and nonarithmetic negatively curved space forms
6.6.2.4 volumes of negatively curved space forms
6.6.3 rank 1 symmetric spaces
6.6.4 higher rank symmetric spaces
6.6.4.1 superrigidity
6.6.5 homogeneous spaces
7 volumes and inequalities on volumes of cycles
7.1 curvature inequalities
7.1.1 bounds on volume elements and first applications.
7.1.1.1 the canonical measure
7.1.1.2 volumes of standard spaces
7.1.1.3 the isoperimetric inequality for spheres
7.1.1.4 sectional curvature upper bounds
7.1.1.5 ricci curvature lower bounds
7.1.2 isoperimetric profile
7.1.2.1 definition and examples
7.1.2.2 the gromov-berard-besson-gallot bound.
7.1.2.3 nonpositive curvature on noncompaet manifolds
7.2 curvature free inequalities on volumes of cycles
7.2.1 curves in surfaces
7.2.1.1 loewner, pu and blatter-bavard theorems
7.2.1.2 higher genus surfaces
7.2.1.3 the sphere
7.2.1.4 homolngical systoles
7.2.2 inequalities for curves
7.2.2.1 the problem, and standard manifolds
7.2.2.2 filling volume and filling radius
7.2.2.3 gromov's theorem and sketch of the proof
7.2.3 higher dimensional systoles:systolic freedom almost everywhere
7.2.4 embolic inequalities
7.2.4.1 introduction
7.2.4.2 the unit tangent bundle
7.2.4.3 the core of the proof
7.2.4.4 croke's three results
7.2.4.5 infinite injectivity radius
7.2.4.6 using embolic inequalities
8 transition: the next two chapters
8.1 spectral geometry and geodesic dynamics
8.2 why are riemannian manifolds so important?
8.3 positive versus negative curvature
9 spectrum of the laplacian
9.1 history
9.2 motivation
9.3 setting up
9.3.1 xdefinition
9.3.2 the hodge star
9.3.3 facts
9.3.4 heat, wave and schrodinger equations
9.4 minimax
9.4.1 the principle
9.4.2 an application
9.5 some extreme examples
9.5.1 square tori, alias several variable fourier series
9.5.2 other flat tori
9.5.3 spheres
9.5.4 kpn
9.5.5 other space forms
9.6 current questions
9.6.1 direct questions about the spectrum
9.6.2 direct problems about the eigenfunctions
9.6.3 inverse problems on the spectrum
9.7 first tools: the heat kernel and heat equation
9.7.1 the main result
9.7.2 great hopes
9.7.3 the heat kernel and ricci curvature
9.8 the wave equation: the gaps
9.9 the wave equation: spectrum & geodesic flow
9.10 the first eigenvalue
9.10.1 λ1 and ricci curvature
9.10.2 cheeger's constant
9.10.3 λ1 and volume; surfaces and multiplicity
9.10.4 kahder manifolds
9.11 results on eigenfunctions
9.11.1 distribution of the eigenfunctions
9.11.2 volume of the nodal hypersurfaces
9.11.3 distribution of the nodal hypersurfaces
9.12 inverse problems
9.12.1 the nature of the image
9.12.2 inverse problems: nonuniqueneas
9.12.3 inverse problems: finiteness, compactness
9.12.4 uniqueness and rigidity results
9.12.4.1 vigneras surfaces
9.13 special cases
9.13.1 riemann surfaces
9.13.2 space forms
9.13.2.1 scars
9.14 the spectrum of exterior differential forms
10 geodesic dynamics
10.1 introduction
10.2 some well understood examples
10.2.1 surfaces of revolution
10.2.1.1 zou surfaces
10.2.1.2 weinstein surfaces
10.2.2 ellipsoids and morse theory
10.2.3 flat and other tori: influence of the fundamental group
10.2.3.1 flat tori
10.2.3.2 manifolds which are not simply connected
10.2.3.3 tori, not flat
10.2.4 space forms
10.2.4.1 space form surfaces
10.2.4.2 higher dimensional space forms
10.3 geodesics joining two points
10.3.1 birkhoff's proof for the sphere
10.3.2 morse theory
10.3.3 discoveries of morse and serre
10.3.4 computing with entropy
10.3.5 rational homology and gromov's work
10.4 periodic geodesics
10.4.1 the difficulties
10.4.2 general results
10.4.2.1 gromoll and meyer
10.4.2.2 results for the generic ("bumpy") case.
10.4.3 surfaces
10.4.3.1 the lusternik-schnirelmann theorem
10.4.3.2 the bangert-franks-hingston results
10.5 the geodesic flow
10.5.1 review of ergodic theory of dynamical systems
10.5.1.1 ergodicity and mixing
10.5.1.2 notions of entropy
10.6 negative curvature
10.6.1 distribution of geodesics
10.6.2 distribution of periodic geodesics
10.7 nonpositive curvature
10.8 entropies on various space forms
10.8.1 liouville entropy
10.9 from osserman to lohkamp
10.10 manifolds all of whose geodesics are closed
10.10.1 definitions and caution
10.10.2 bott and samelson theorems
10.10.3 the structure on a given sa and ki~
10.11 inverse problems: conjngacy of geodesic flows
11 best metrics
11.1 introduction and a possible approach
11.1.1 an approach
11.2 purely geometric functionals
11.2.1 systolic inequalities
11.2.2 counting periodic geodesics
11.2.3 the embolic constant
11.2.4 diameter and injectivity
11.3 least curved
11.3.1 definitions
11.3.1.1 inf
· · · · · · (收起)
1 euclidean geometry
1.1 preliminaries
1.2 distance geometry
1.2.1 a basic formula
1.2.2 the length of a path
1.2.3 the first variation formula and application to billiards
1.3 plane curves
1.3.1 length
1.3.2 curvature
1.4 global theory of closed plane curves
1.4.1 "obvious" truths about curves which are hard to prove
1.4.2 the four vertex theorem
1.4.3 convexity with respect to arc length
1.4.4 umlaufsatz with corners
1.4.5 heat shrinking of plane curves
1.4.6 arnol'd's revolution in plane curve theory
1.5 the isoperimetric inequality for curves
1.6 the geometry of surfaces before and after gaul}
1.6.1 inner geometry: a first attempt
.1.6.2 looking for shortest curves: geodesics
1.6.3 the second fundamental form and principal curvatures
1.6.4 the meaning of the sign of k
1.6.5 global surface geometry
1.6.6 minimal surfaces
1.6.7 the hartman-nirenberg theorem for inner flat surfaces
1.6.8 the isoperimetric inequality in e3 a la gromov
1.6.8.1 notes
1.7 generic surfaces
1.8 heat and wave analysis in e2
1.8.1 planar physics
1.8.1.1 bibliographical note
1.8.2 why the eigenvalue problem?
1.8.3 minimax
1.8.4 shape of a drum
1.8.4.1 a few direct problems
1.8.4.2 the faber-krahn inequality
1.8.4.3 inverse problems
1.8.5 heat
1.8.5.1 eigenfunctions
1.8.6 relations between the two spectra
1.9 heat and waves in e3, ed and on the sphere
1.9.1 euclidean spaces
1.9.2 spheres
1.9.3 billiards in higher dimensions
1.9.4 the wave equation versus the heat equation
2 transition
3 surfaces from gaufi to today
3.1 gau6
3.1.1 theorema egregium
3.1.1.1 the first proof of ganβ's theorema egregium; the concept of ds2
3.1.1.2 second proof of the theorema egregium.
3.1.2 the gang-bonnet formula and the rodrigues-gauiβ map
3.1.3 parallel transport
3.1.4 inner geometry
3.2 alexandrov's theorems
3.2.1 angle corrections of legendre and gauβ in geodesy
3.3 cut loci
3.4 global surface theory
3.4.1 bending surfaces
3.4.1.1 bending polyhedra
3.4.1.2 bending and wrinkling with little smoothness
3.4.2 mean curvature rigidity of the sphere
3.4.3 negatively curved surfaces
3.4.4 the willmore conjecture
3.4.5 the global gautβ-bonnet theorem for surfaces
3.4.6 the hopf index formula
4 riemann's blueprints
4.1 smooth manifolds
4.1.1 introduction
4.1.2 the need for abstract manifolds
4.1.3 examples
4.1.3.1 submanifolds
4.1.3.2 products
4.1.3.3 lie groups
4.1.3.4 homogeneous spaces
4.1.3.5 orassmannians over various algebras
4.1.3.6 gluing
4.1.4 the classification of manifolds
4.1.4.1 surfaces
4.1.4.2 higher dimensions
4.1.4.3 embedding manifolds in euclidean space
4.2 calculus on manifolds
4.2.1 tangent spaces and the tangent bundle
4.2.2 differential forms and exterior calculus
4.3 examples of riemann's definition
4.3.1 riemann's definition
4.3.2 hyperbolic geometry
4.3.3 products, coverings and quotients
4.3.3.1 products
4.3.3.2 coverings
4.3.4 homogeneous spaces
4.3.5 symmetric spaces
4.3.5.1 classification
4.3.5.2 rank
4.3.6 riemannian submersions
4.3.7 gluing and surgery
4.3.7.1 gluing of hyperbolic surfaces
4.3.7.2 higher dimensional gluing
4.3.8 classical mechanics
4.4 the riemann curvature tensor
4.4.1 discovery and definition
4.4.2 the sectional curvature
4.4.3 standard examples
4.4.3.1 constant sectional curvature
4.4.3.2 projective spaces kpn
4.4.3.3 products
4.4.3.4 homogeneous spaces
4.4.3.5 hypersurfaces in euclidean space
4.5 a naive question: does the curvature determine the metric?
4.5.1 surfaces
4.5.2 any dimension
4.6 abstract riemannian manifolds
4.6.1 isometrically embedding surfaces in e3
4.6.2 local isometric embedding of surfaces in ms
4.6.3 isometric embedding in higher dimensions
5 a one page panorama
6 metric geometry and curvature
6.1 first metric properties
6.1.1 local properties
6.1.2 hopf-rinow and de rham theorems
6.1.2.1 products
6.1.3 convexity and small balls
6.1.4 totally geodesic submanifolds
6.1.5 center of mass
6.1.6 examples of geodesics
6.1.7 transition
6.2 first technical tools
6.3 second technical tools
6.3.1 exponential map
6.3.1.1 rank
6.3.2 space forms
6.3.3 nonpositive curvature
6.4 triangle comparison theorems
6.4.1 bounded sectional curvature
6.4.2 ricci lower bound
6.4.3 philosophy behind these bounds
6.5 injectivity, convexity radius and cut locus
6.5.1 definition of cut points and injectivity radius
6.5.2 klingenberg and cheeger theorems
6.5.3 convexity radius
6.5.4 cut locus
6.5.5 blaschke manifolds
6.6 geometric hierarchy
6.6.1 the geometric hierarchy
6.6.1.1 space forms
6.6.1.2 rank 1 symmetric spaces
6.6.1.3 measure isotropy
6.6.1.4 symmetric spaces
6.6.1.5 homogeneous spaces
6.6.2 constant sectional curvature
6.6.2.1 negatively curved space forms in three and higher dimensions
6.6.2.2 mostow rigidity
6.6.2.3 classification of arithmetic and nonarithmetic negatively curved space forms
6.6.2.4 volumes of negatively curved space forms
6.6.3 rank 1 symmetric spaces
6.6.4 higher rank symmetric spaces
6.6.4.1 superrigidity
6.6.5 homogeneous spaces
7 volumes and inequalities on volumes of cycles
7.1 curvature inequalities
7.1.1 bounds on volume elements and first applications.
7.1.1.1 the canonical measure
7.1.1.2 volumes of standard spaces
7.1.1.3 the isoperimetric inequality for spheres
7.1.1.4 sectional curvature upper bounds
7.1.1.5 ricci curvature lower bounds
7.1.2 isoperimetric profile
7.1.2.1 definition and examples
7.1.2.2 the gromov-berard-besson-gallot bound.
7.1.2.3 nonpositive curvature on noncompaet manifolds
7.2 curvature free inequalities on volumes of cycles
7.2.1 curves in surfaces
7.2.1.1 loewner, pu and blatter-bavard theorems
7.2.1.2 higher genus surfaces
7.2.1.3 the sphere
7.2.1.4 homolngical systoles
7.2.2 inequalities for curves
7.2.2.1 the problem, and standard manifolds
7.2.2.2 filling volume and filling radius
7.2.2.3 gromov's theorem and sketch of the proof
7.2.3 higher dimensional systoles:systolic freedom almost everywhere
7.2.4 embolic inequalities
7.2.4.1 introduction
7.2.4.2 the unit tangent bundle
7.2.4.3 the core of the proof
7.2.4.4 croke's three results
7.2.4.5 infinite injectivity radius
7.2.4.6 using embolic inequalities
8 transition: the next two chapters
8.1 spectral geometry and geodesic dynamics
8.2 why are riemannian manifolds so important?
8.3 positive versus negative curvature
9 spectrum of the laplacian
9.1 history
9.2 motivation
9.3 setting up
9.3.1 xdefinition
9.3.2 the hodge star
9.3.3 facts
9.3.4 heat, wave and schrodinger equations
9.4 minimax
9.4.1 the principle
9.4.2 an application
9.5 some extreme examples
9.5.1 square tori, alias several variable fourier series
9.5.2 other flat tori
9.5.3 spheres
9.5.4 kpn
9.5.5 other space forms
9.6 current questions
9.6.1 direct questions about the spectrum
9.6.2 direct problems about the eigenfunctions
9.6.3 inverse problems on the spectrum
9.7 first tools: the heat kernel and heat equation
9.7.1 the main result
9.7.2 great hopes
9.7.3 the heat kernel and ricci curvature
9.8 the wave equation: the gaps
9.9 the wave equation: spectrum & geodesic flow
9.10 the first eigenvalue
9.10.1 λ1 and ricci curvature
9.10.2 cheeger's constant
9.10.3 λ1 and volume; surfaces and multiplicity
9.10.4 kahder manifolds
9.11 results on eigenfunctions
9.11.1 distribution of the eigenfunctions
9.11.2 volume of the nodal hypersurfaces
9.11.3 distribution of the nodal hypersurfaces
9.12 inverse problems
9.12.1 the nature of the image
9.12.2 inverse problems: nonuniqueneas
9.12.3 inverse problems: finiteness, compactness
9.12.4 uniqueness and rigidity results
9.12.4.1 vigneras surfaces
9.13 special cases
9.13.1 riemann surfaces
9.13.2 space forms
9.13.2.1 scars
9.14 the spectrum of exterior differential forms
10 geodesic dynamics
10.1 introduction
10.2 some well understood examples
10.2.1 surfaces of revolution
10.2.1.1 zou surfaces
10.2.1.2 weinstein surfaces
10.2.2 ellipsoids and morse theory
10.2.3 flat and other tori: influence of the fundamental group
10.2.3.1 flat tori
10.2.3.2 manifolds which are not simply connected
10.2.3.3 tori, not flat
10.2.4 space forms
10.2.4.1 space form surfaces
10.2.4.2 higher dimensional space forms
10.3 geodesics joining two points
10.3.1 birkhoff's proof for the sphere
10.3.2 morse theory
10.3.3 discoveries of morse and serre
10.3.4 computing with entropy
10.3.5 rational homology and gromov's work
10.4 periodic geodesics
10.4.1 the difficulties
10.4.2 general results
10.4.2.1 gromoll and meyer
10.4.2.2 results for the generic ("bumpy") case.
10.4.3 surfaces
10.4.3.1 the lusternik-schnirelmann theorem
10.4.3.2 the bangert-franks-hingston results
10.5 the geodesic flow
10.5.1 review of ergodic theory of dynamical systems
10.5.1.1 ergodicity and mixing
10.5.1.2 notions of entropy
10.6 negative curvature
10.6.1 distribution of geodesics
10.6.2 distribution of periodic geodesics
10.7 nonpositive curvature
10.8 entropies on various space forms
10.8.1 liouville entropy
10.9 from osserman to lohkamp
10.10 manifolds all of whose geodesics are closed
10.10.1 definitions and caution
10.10.2 bott and samelson theorems
10.10.3 the structure on a given sa and ki~
10.11 inverse problems: conjngacy of geodesic flows
11 best metrics
11.1 introduction and a possible approach
11.1.1 an approach
11.2 purely geometric functionals
11.2.1 systolic inequalities
11.2.2 counting periodic geodesics
11.2.3 the embolic constant
11.2.4 diameter and injectivity
11.3 least curved
11.3.1 definitions
11.3.1.1 inf
· · · · · · (收起)
读后感
我当然没有读完这本书，所以这不是一篇书评。事实上，我是今天才买到这本书的。 离开学术的日子，偶尔去书店时，还依然会条件反射般先跑到原版数学书的货架前，翻一翻新出的旧书（没错，事实正是如此）。若在从前，可能还会YY一下若是能啃下来这一本，内功修为又可以增长几何...
我当然没有读完这本书，所以这不是一篇书评。事实上，我是今天才买到这本书的。 离开学术的日子，偶尔去书店时，还依然会条件反射般先跑到原版数学书的货架前，翻一翻新出的旧书（没错，事实正是如此）。若在从前，可能还会YY一下若是能啃下来这一本，内功修为又可以增长几何...
我当然没有读完这本书，所以这不是一篇书评。事实上，我是今天才买到这本书的。 离开学术的日子，偶尔去书店时，还依然会条件反射般先跑到原版数学书的货架前，翻一翻新出的旧书（没错，事实正是如此）。若在从前，可能还会YY一下若是能啃下来这一本，内功修为又可以增长几何...
我当然没有读完这本书，所以这不是一篇书评。事实上，我是今天才买到这本书的。 离开学术的日子，偶尔去书店时，还依然会条件反射般先跑到原版数学书的货架前，翻一翻新出的旧书（没错，事实正是如此）。若在从前，可能还会YY一下若是能啃下来这一本，内功修为又可以增长几何...
我当然没有读完这本书，所以这不是一篇书评。事实上，我是今天才买到这本书的。 离开学术的日子，偶尔去书店时，还依然会条件反射般先跑到原版数学书的货架前，翻一翻新出的旧书（没错，事实正是如此）。若在从前，可能还会YY一下若是能啃下来这一本，内功修为又可以增长几何...
用户评价
这本书的章节安排极具匠心，特别是关于流形和张量分析的引入部分，设计得非常巧妙。作者首先从欧几里得空间中的曲线和曲面入手，用最直观的方式建立了局部坐标系和第一基本形式的概念，然后水到渠成地过渡到了抽象的黎曼流形。这种“由具体到抽象”的教学策略，有效地降低了读者面对高维抽象概念时的心理门槛。我特别欣赏它对“测地线”的阐述，通过对惯性运动的几何解释，让原本复杂的变分问题变得清晰可感。总而言之，这本书成功地搭建了一座坚实的桥梁，连接了直观的几何想象与严谨的现代数学语言，为读者开启了一扇通往深奥空间理论的大门，读完后感觉整个认知地图都被拓展了。
这本书的封面设计非常有品味，那种深沉的蓝色调搭配简约的字体，一看就知道这是一本严肃的学术著作，而不是那种轻飘飘的入门读物。我最初被它吸引，是因为我对数学的某些抽象概念一直抱有浓厚的兴趣，尤其是那些涉及到空间结构和拓扑特性的理论。拿到手里的时候，感觉分量十足，那种沉甸甸的感觉让人对内容的深度充满了期待。我翻开扉页，看到作者的简介，确实是该领域内的权威人物，这更增加了我对这本书的信心。我期望它能为我揭开那些隐藏在复杂公式背后的几何直觉，让我能够真正“看”到那些高维空间是如何运作的。我不是科班出身，所以对这类书籍的阅读体验要求较高，它既需要有严谨的逻辑推导，又不能让初学者感到完全的不知所措，希望这本书能在两者之间找到一个完美的平衡点。
说实话，这本书的难度曲线相当陡峭，它绝不是一本可以轻松“翻阅”的书籍。我花了大量时间去消化其中的每一个定理和证明的细节。有些章节的证明过程极其精巧，需要读者具备扎实的微积分和线性代数基础，才能跟上作者的思维步伐。我发现自己经常需要停下来，拿起草稿纸，自己重新推导一遍，以确保完全理解了每一步逻辑的必然性。虽然过程有些煎熬，但每当攻克一个难点，那种豁然开朗的成就感是无与伦比的。这本书的价值恰恰在于其对严谨性的坚持，它没有为了迎合大众而简化核心内容，而是忠实地呈现了该领域最本质的数学结构。对于想要深入研究，甚至未来准备从事相关领域研究的人来说，这无疑是一部不可或缺的“案头工具书”。
这本书的排版堪称教科书级别的典范，字体大小适中，行距宽松，使得在长时间阅读后眼睛的疲劳感也相对较轻。最让我赞赏的是，插图和图表的质量非常高，它们不仅仅是作为公式的辅助说明，更是理解抽象概念的关键桥梁。许多涉及微分形式和曲率张量的描述，如果没有那些清晰的几何图示，我恐怕会陷入无尽的符号迷宫中。作者似乎非常理解读者在学习这些高深理论时会遇到的困难点，总能在关键的转折处加入详尽的几何直觉铺垫，而不是直接跳跃到纯粹的代数操作。这对我这样试图从几何图像去理解代数结构的学习者来说，简直是雪中送炭。我特别喜欢它对经典微分几何概念的引入方式，那种循序渐进的感觉，让人觉得复杂的理论体系并非遥不可及。
从图书馆借阅的这本厚重的书，散发出一种独特的、混合了纸张和油墨的学术气息。我注意到书中似乎收录了许多历史上重要的里程碑式的思想片段，不仅仅是现代的框架。作者在回顾理论发展脉络时，展现了一种深厚的历史洞察力，这使得整个学习过程不仅仅是知识的灌输，更像是一场数学思想的朝圣之旅。例如，他对一些早期微分几何学家如何面对“非欧几里得”空间挑战的描述，充满了人文关怀和智力上的敬意。这让我感受到，数学并非是冰冷无情的逻辑堆砌，而是一系列人类伟大智慧探索的结晶。这种兼顾理论深度与历史纵深的叙事方式，极大地提升了阅读体验，让我对纯粹的数学美学有了更深层次的体会。
数学还有很多问题，
在文轩网下的单，刚拿到书，大失所望，印刷质量很差，纸张有点发霉，而且闻起来一股味道。。。与我在新街口新华书店看到的那个版本完全不能比。感觉是盗印的，Fuck
数学还有很多问题，
在文轩网下的单，刚拿到书，大失所望，印刷质量很差，纸张有点发霉，而且闻起来一股味道。。。与我在新街口新华书店看到的那个版本完全不能比。感觉是盗印的，Fuck
在文轩网下的单，刚拿到书，大失所望，印刷质量很差，纸张有点发霉，而且闻起来一股味道。。。与我在新街口新华书店看到的那个版本完全不能比。感觉是盗印的，Fuck
