古典微分几何

1， Descartes坐标系、坐标变换、Euclid空间中的曲线、梯度、余向量、Riemann度量、伪Riemann度量、Minkowski度量。

1。光滑曲线，参数化，切线，法线。

2。光滑曲面。

3。曲面的坐标，坐标曲线，光滑曲面的几何，切向量，内蕴坐标。

4。曲面间的映射，坐标变换，微分同胚的概念，局部坐标基变换的雅可比矩阵。

5。切平面及其方程，切平面间的距离。

6。曲线的弧长，自然参数。

7。曲线的曲率，密切面与密切圆，平缓曲线的曲率。

2， 正则曲线与Frenet三角形与Frenet挠曲线，平面曲线、具有常曲率的平面曲线、空间曲线、曲率与挠率的关系。

3， Frenet方程、Frenet公式。

9。扭转定理，Frenet三角形的扭转。

局部曲线论的基本定理、Minkowski空间、Minkowski空间上的Frenet方程、闭曲线、缠绕数、旋转度、凸曲线及其分类、四顶点定理。

10。曲率与挠率的计算公式。

11。曲面与空间曲线的自然参数方程。

12。曲面的第一基本形式，切向量的长度和夹角，内蕴坐标下的曲面面积，曲面分类问题的不同方法，

13。曲面上曲线的曲率与曲绿中心，Meusnier定理。

4， 狭义相对论的数学模型、Poincare群、Lorenz变换、曲面元、曲面的第一基本形式、曲面的定向、曲面上的诱导度量。

14。曲面的曲率与曲率中心，主曲率与主方向，曲面的第二基本形式。

5， Gauss映射、Weingarten映射、曲面的第二与第三基本形式、

15。作为第二基本形式下不变量的主曲率与主方向，欧拉公式，高斯曲率及其几何意义。

主曲率、主曲率与主方向的计算公式。旋转面、Beltrami-Enneper定理、直纹面。

6， 可展曲面、Weingarten曲面、极小曲面、共形参数化。

17。欧氏空间中曲线坐标下的光滑曲线与切向量，坐标变换，微分同胚，维数不变性，局部坐标基变换的雅可比矩阵。

18。欧氏空间中曲面的活动坐标，坐标变换与局部基，维数不变性，局部连续，活动坐标系下的方程组。

19。曲线坐标中的欧几里德度量，弧长、曲线间角度、体积，极坐标、柱坐标、球坐标下的映射。

20。黎曼度量及其例子，弧长、曲线间的角度、体积，等距映射，与欧几里德度量的等价。

7， Weierstrass表示、Minkowski空间上的曲面、超曲面、球面上的度量。

21。伪欧氏空间，正交完备性，正交基。

22。伪正交基与变换。

23。伪正交平面，解析表示，伪正交平面上的向量的角度。

24。欧氏空间上的正交群，伪正交群的结构。

25。二维球面与伪球面的几何，三角不等式。

26。球面与伪球面的唯一性，变换群作用下的球面与伪球面。

27。作为罗巴切夫斯基平面的伪球面，罗巴切夫斯基平面的凯莱模型及其变换群，欧几里德第五公设的独立性，欧几里德、黎曼与罗巴切夫斯基几何的概念。

28。极坐标下罗巴切夫斯基平面与球上的度量，闭曲线的长度与面积。

29。射影坐标下罗巴切夫斯基平面与球上的度量。

30。旋转面的坐标及主曲率，旋转面上的罗巴切夫斯基平面上的曲线及其曲率。

31。共形欧氏度量与等距坐标，球面与罗巴切夫斯基平面上的三角形的内角和的估计，等价度量。

8， Lobachevsky度量、Lobachevsky几何的Poincare度量模型与Klein度量模型、Minkowski空间中的类空曲面的曲率、复变换群、复解析函数、Riemann曲面、共形坐标。

9， Beltrami方程、球面度量与Lobachevsky度量、常曲率空间、矩阵空间中的曲面、矩阵的指数映射。

32。向量值函数的导数，仿射空间上的可微向量场及其基本性质。

10， 四元数、共形度量、共形变换、Liouville定理、向量场的可微性，方向导数、共变导数、协变微分与内蕴微分及其基本性质。

联络、曲面内蕴坐标上的微分算子，Christoffel符号、

35。Christoffel符号的对称性，Christoffel恒等式。

Gauss公式、Weingarten方程。

11， 平行向量场、曲面上曲线的测地曲率，测地线、

37。测地线的方程，测地线的存在与唯一性，罗巴切夫斯基平面上的测地线。

38。通过两点的测地线，测地半径。

39。测地距，于紧集上极限的关系。

40。二维曲面上的半测地坐标。

41。二维曲面上作为最短距离的测地线。

平行移动、

42。等距曲线的平行移动，向量场导数的行列式。

43。曲面上等距曲线向量场的旋转、转速和转角。

44。有界曲面上向量间的角度与平行移动，向量场的旋转与测地三角形的内角和。

45。有界曲面上向量间的角度与平行移动及曲面曲率之间的关系。

最短路径定理、高斯曲率的不变性，Gauss绝妙定理、Gauss方程、Codazzi-Mainardi 方程、曲率张量、局部曲面论的基本定理、Gauss曲率、测地平行坐标。

12， 曲面的同构、Maurer-Cartan方程、测地曲率、Gauss-Bonnet定理。球面与罗巴切夫斯基平面上的三角形的内角。

13， 曲面的大范围性质、Riemann与伪Riemann空间中的张量、伪微分同胚的单参数群、向量场的指数映射。

47。曲面上的球面映射。

48。曲面上的复结构，球面上的复结构，复形式的共形欧几里德度量。

Differential Geometry

1，B.A.Dubrovin、A.T.Fomenko、S.P.Novikov，现代几何学。

2，P.K.Rachevsky，微分几何教程，第13-58节（除23、29、30、33、43节）及第86-88节。

3，S.P.Novikov、A.T.Fomenko，微分几何与拓扑学初步，第一部分。

4，A.S.Mishchenko、A.T.Fomenko，微分几何与拓扑学简明教程，第1章第1、2节、第2章第4节、第4章、第5章。

5，P.K.Rachevsky，黎曼几何与张量解析，第44-48节。

A?C?菲金科《微分几何习题集》北京师范大学出版社

《微分几何理论与习题》里普希茨

A.T.Fomenko Differential geometry and topology，Consultants Bureau
Dubrovin, Fomenko, Novikov “Modern geometry-methods and applications”Vol 1—3
A Comprehensive Introduction to Differential Geometry vol 1-5 ,by Michael Spivak 

Differential Geometry of Curves and Surfaces by Manfredo Do Carmo
M. Postnikov，Linear algebra and differential geometry，Mir Publishers

W.M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry Academic Press, Inc., Orlando, FL, 1986

Chen Qing and Chia Kuai Peng, Differential Geometry  

Eisenhart的”Diffenrential Geometry(?)” 

N. Hicks, Notes on differential geometry, Van Nostrand.

Hilbert ,foundations of geometry；

T. Frenkel, Geometry of Physics

Peter Petersen, Riemannian Geometry：

Riemannian Manifolds: An Introduction to Curvature by John M. Lee：

Helgason , Differential Geometry,Lie groups,and symmetric spaces：

Lang, Fundamentals of Differential Geometry：

kobayashi/nomizu, Foundations of Differential Geometry：

Riemannian Geometry I.Chavel：

Darboux的”Lecons sur la theorie generale des surfaces”。

Gauss的”Disquisitiones generales circa superficies curvas”。

P.Dombrowski的”150 years after Gauss‘ ‘Disquisitiones generales circa superficies curvas‘ “

R.Osserman的”Lectures of Minimal Surfaces”

J.C.C.Nitsche的”Lectures on Minimal Surfaces”(Vol.1) 

陈省身，《微分几何讲义》，北大版
陈维桓，《微分流形初步》，高教版
苏步青, 《微分几何》，高教版

吴大任的”微分几何学(?)”,《微分几何讲义》高等教育出版社

沈纯理,黄宣国的”微分几何”(经济科学出版社,97)。

姜国英,黄宣国的”微分几何100例”。

彭家贵《微分几何》高等教育出版社

陈省身《微分几何》南开大学讲义

178《微分几何》第4版 梅向明, 黄敬之编 

181《微分几何》周建伟著

185《微分几何讲义》吴大任

【习题集与辅导书】

187《微分几何习题集》杨文茂,傅朝金,程新跃编著

188《微分几何理论与习题》里普希茨

189《微分几何学习指导与习题选解》梅向明，王汇淳编

【提高】

191《微分几何五讲》苏步青著

192《微分几何讲义》丘成桐，孙理察著

193《微分几何入门与广义相对论》梁灿彬，周彬著

Differential Forms

X

(Geometry of) Manifolds

Lang, Differential and Riemannian manifolds：

Warner,Foundations of Differentiable manifolds and Lie groups：

Introduction to Smooth Manifolds by John M. Lee：

1.W.M.Boothby “An Introduction to Differentiable Manifolds and Riemannian Geometry” 

B.A. Dubrovin, A.T. Fomenko, S.P. Novikov “Modern Geometry–Methods and Applications”的第一,二卷

Gallot, Hulin, Lafontain “Introduction to Riemannian Geometry”(?) 

J.Milnor Topology from a differential point of view (中译本:从微分观点看拓扑)

J.Milnor Morse Theory (中译本:莫尔斯理论)讲

Spivak “Calculus on Manifolds”(?) (中文名字就叫”流形上的微积分”).

V.I.Arnold “Mathematical Mathods of Classical Mechanics”

R.Narasimhan “Analysis on Real and Complex Manifolds” 

C. von Westenholz “Differential forms in Mthematical Physics” 

陈省身,陈维桓的”微分几何初步” 

白正国,沈一兵,水乃翔,郭效英 “黎曼几何初步”。

苏竞存 “流形的拓扑学”. 此书块头很大,内容翔实,而且有很多作者加的话, 有意思. 有本书,可能不入高手法眼,不过我觉得是很不错的,

Bott, Raoul, R. Bott, and Loring W. Tu. Differential Forms in Algebraic Topology (Graduate Texts in Mathematics; 82). Reprint edition. New York: Springer-Verlag, June 1, 1995. ISBN: 0387906134.

Buy at Amazon Abraham, Ralph, Jerrold E. Marsden, and Tudor Ratiu. “Manifolds, Tensor Analysis, and Applications.” Applied Mathematical Sciences. Vol. 75. New York: Springer Verlag, May 1, 1998. ISBN: 3540967907.

Buy at Amazon Hirsch, Morris W. “Differential Topology.” Graduate Texts in Mathematics. Vol. 33. Reprint edition. New York: Springer-Verlag, November 1, 1988. ISBN: 0387901485.

Geometric Analysis

Peter Li

Yau

微分几何与拓扑学

4。切向量与超曲面的切平面，可微函数的曲率张量。

5。流形上的光滑映射的可微形。

1， 向量场、光滑向量场和他的积分曲线，流、管状邻域、

7。流形的切向量丛，光滑向量丛。

8。正则映射下流形的逆映射的结构。

9。正则映射下紧致光滑流形的逆映射上的向量丛。

10。向量场的换位子，它的性质。

11。全纯向量丛，它的性质。

纤维丛、向量丛、球丛、拓扑群、轨道空间。

2， 透镜空间、同伦、同伦的映射、同伦类、基本群、基本群的运算、道路提升引理、同伦提升引理、轨道空间的基本群、乘积空间的基本群。

3， 同伦型、形变收缩、可缩空间、Brouwer不动点定理、Jordan曲线定理、曲面的边界、单纯形、单纯剖分、单纯复形、可单纯剖分空间、重心重分。

4， 承载形、单纯逼近定理、复形的棱道群、Van Kampen定理、轨道空间的单纯剖分、无穷复形。

5， 闭曲面的分类、曲面的可定向性、Euler示性数、曲面的符号、亏格。

1。有限覆盖，单位分解定理。

2。紧致流形嵌入Euclid空间。

12。线性微分形式，一些基本的线性微分形式。

13。张量场，局部性质。

14。张量的加法，多重线性函数，张量积，基本的张量场。

15。缩并运算，例子。

16。对称与斜对称张量，交错对称算子。

19。斜对称张量空间的基本信息，微分形式的基与坐标。

20。微分形式的坐标变换与坐标变换下的基的映射。

21。有向流形，有向图册，有向微分形式。

22。外微分，坐标下的外微分运算。

23。三维空间向量场的外微分运算。

24。de Rham上同调，de Rham定理。

25。de Rham上同调与流形上的微分形式的光滑映射的作用。

26。de Rham上同调的同伦性质。

27。de Rham上同调的同伦不变性，Poincare定理。

6， Riemann度量、Riemann流形、具有Riemann度量的光滑流形。Riemann乘积流形、Riemann子流形、Riemann浸没、复射影空间、齐性Riemann空间、Steenrod定理、联络、Levi-Civita联络、Riemann子流形的联络。

28。流形与子流形上微分形式的积分。

29。Riemann流形上函数的积分及其与微分形式的积分的比较，与三维空间上曲面与曲线上的向量场的积分的比较。

30。一般Stokes公式，特殊形式及结果（三维空间上的向量场、de Rham上同调）。

31。Euclid空间上向量场的微分，及其基本性质。

32。仿射的联络，向量场的绝对微分，坐标形式，Christoffel符号，等价联络，仿射等价联络。

33。对称仿射联络。

34。纵向曲线向量的微分与平行移动，切空间的变换，向量场上的微分算子的性质。

35。曲线沿着余向量与任意张量的移动，张量积与旋度。

36。张量的协变微分。

37。向量函数与张量场次微分的坐标形式，平行条件，张量的梯度，向量场的分叉。

38。Riemann流形上仿射联络的协调性条件。

39。Riemann流形上仿射联络存在与唯一性的Levi-Civita定理。

40。欧氏空间、伪欧氏空间与Riemann流形上的子流形的协变微分与仿射联络。

41。测地线及其方程，测地线方程的存在性，指定方向上某点的测地线的存在性与唯一性。

42。Riemann流形上的曲率，Riemann流形的子流形上的曲线的测地曲率，球上与Lobachevsky平面上的曲率。

7， 沿曲线的共变导数、平行移动、测地线、测地线的局部存在性与唯一性、指数映射、Gauss引理、完备Riemann流形、Hopf-Rinow定理。

43。初等变分法：极值的Lagrange与Euler方程。

44。作为作用量与长度泛函的极值的测地线。

45。点的邻域上的测地性质，存在性与可延伸性，测地完备流形上的测地线的无限延伸性。

46。闭集上的连通性与测地线，法坐标，测地球面，测地球面的半径，他们的正交性，测地极值。

47。Riemann流形的曲率张量。

48。曲率章量的性质及其坐标。

49。Riemann度量的符号。

50。二维Riemann流形的曲率，作为零曲率空间的局部欧氏空间。

51。截面曲率作为测地曲面的总曲率。

53。二维Riemann流形上沿着封闭道路的旋转张量场。

54。二维流形上闭曲线的环绕向量，测地三角形的内角和。

55。几何平均曲率张量（闭曲线的环绕向量及其与二重向量之间的关系）。

56。二维流形上沿着给定方向的几何曲率。

8， 割迹、第二共变导数、曲率张量的代数性质、曲率的计算、Ricci曲率、标量曲率、第一变分形式、第二变分形式、Jacobi场、水平提升。

9， O’Neill公式、正规齐性度量、Gauss引理、共轭点、具有常截面曲率的空间、Myers定理、Hadamard定理、微分流形上的可测集、体积估计、有限群的指数增长性、Milnor-Wolf定理。

10， 负曲率紧致流形的基本群的增长性、Milnor定理、Gauss-Bonnet公式、Gromov定理、Cheeger定理、共形平坦流形、第二Bianchi等式、单纯

同调群、边缘闭链、定向单纯形、同调群、同调类、单纯映射、链复形、辐式重分。

11， 映射度、连续向量场、Euler-Poincare公式、有理系数同调群、Borsuk-Ulam定理、Lusternik定理、Lefschetz不动点定理、Hopf定理。

12， 维数、纽结的等价、纽结群、Seifert曲面、覆盖空间、映射提升定理、万有覆盖空间。

13， 链环、Kauffman纽结多项式、Jones纽结多项式、Conway纽结多项式、Alexander纽结多项式、Vassiliev纽结不变量、Kontsevich定理。

1，S.P.Novikov、A.T.Fomenko，微分几何与拓扑学初步，科学出版社，1987。

2，B.A.Dubrovin、S.P.Novikov、A.T.Fomenko，现代几何学，科学出版社，1985。

3，P.K.Rachevsky，黎曼几何与张量解析，技术与理论文献出版社，1953。

4，A.S.Mishchenko、A.T.Fomenko，微分几何与拓扑学教程，法克特里亚出版社，2000。

5，S.P.Novikov、I.A.Taimanov，现代几何结构与场论，莫斯科独立大学出版社，2004。

6，J.Milnor，Morse理论，莫斯科世界图书出版社，1985。

陈维桓，《黎曼几何引论》（上、下册），北大版
伍宏熙，《黎曼几何初步》，北大版

Classical differential geometry

1 , Descartes coordinate system, the coordinate transformation,Euclid space curves, gradients, vector,Riemann metric, pseudo- Riemann metric, Minkowski metric.

1 。 Smooth curve parameterized, tangents, normals.

2 。 Smooth surfaces.

3 。 The coordinates of a surface, coordinate curves, smooth surface geometry, the tangent vector, intrinsic coordinates.

4 。 The mapping between the surfaces, coordinate transformations, diffeomorphisms, the concept of local coordinate basis transformation of Jacobian matrix.

5 。 Tangent plane equations, the distance between the cutting plane.

6 。 The arc length of curves, natural parameter.

7 。 The curvature of a curve, close and the osculating circle, gentle curvature of the curve.

2 , Regular curves and Frenet triangle and Frenet flexible curves , with constant curvature of plane curves, plane curves, space curve, the relationship between curvature and torsion.

3 , Frenet equations,Frenet formulas.

9 。 Torsion theorem,Frenet triangle reversed.

Fundamental theorem of the theory of local curve, Minkowski Space, Minkowski Space on the Frenet Equation, the number of closed curves, winding, rotation, four-vertex theorem, convex curve and its classification.

10 。 Calculation formulas of curvature and torsion.

11 。 Surfaces and natural parameter equation of a space curve.

12 。 The first fundamental form of a surface, tangent vector length and angle, surface area of intrinsic coordinates, classification of surfaces of different methods

13 。 The curvature of curves on the surface and curved Green Center,Meusnier theorem.

4 , Special relativity mathematics model and thePoincare Group,Lorenz transformations, surface, surface of the first fundamental form and surface orientation, the induced metric on the surface.

14 。 The center of curvature and the curvature of the surface, the principal curvatures and directions, the second fundamental form of a surface.

5 , Gauss map,Weingarten map, surface, the second and the third fundamental form,

15 。 As the second basic form non-variable curvature and direction, Euler’s formula, Gaussian curvature and its geometrical significance.

Principal curvature, Formula for calculating the principal curvatures and principal directions. Rotating surface,Beltrami-Enneper theorem, ruled.

6 , Developable surfaces,Weingarten surfaces, minimal surfaces, Conformal parameterization.

17 。 Curvilinear coordinates in the Euclidean space of smooth curves and tangent vectors, coordinate transformations, diffeomorphisms, invariance of dimension, the local Jacobian matrix of the coordinate basis transformation.

18 。 Activity coordinates of a surface in Euclidean space, coordinate transformations and local base, invariance of dimension, local continuous equations of the active coordinate system.

19 。 Curvilinear coordinates in the Euclidean metric, arc length, angles between curves, volumes, polar, cylindrical, spherical coordinates mapping.

20 。 Riemannian metrics and examples, arc length, the angle between curves, volumes, isometries, equivalent to the Euclidean metric.

7 , Weierstrass representation,Minkowski space on the surface, hypersurface, spherical measure on.

21 。 Pseudo-Euclidean space of orthogonal complete, orthogonal basis.

22 。 Pseudo-orthogonal basis transform.

23 。 Pseudo orthogonal plane, analytic expression for angle of pseudo orthogonal vectors in the plane.

24 。 Orthogonal groups on the Euclidean space, structure of pseudo orthogonal group.

25 。 Two-dimensional sphere and pseudosphere geometry, the triangle inequality.

26 。 The uniqueness of the sphere and pseudosphere, sphere and pseudosphere under transformation groups.

27 。 As Lobachevsky plane, pseudosphere, Gloria of the Lobachevsky plane models and transformation group, Euclid’s five postulates of independence, Euclid, Riemannian and lobachevskian geometry concepts.

28 。 Measurement on the Lobachevsky plane in polar coordinates with the ball closed curve of length and area.

29 。 Measurement on the Lobachevsky plane projective coordinates with the ball.

30 。 Rotating the coordinates and the principal curvatures of the surface, and rotation on the surface of luobaqiefusijiping curves and curvatures of the surface.

31 。 Conformal Euclidean metric and raster coordinates, the sum of the Interior angles of a triangle on a sphere and luobaqiefusijiping estimated that equivalent measures.

8 , Lobachevsky metric,Lobachevsky geometry of the Poincare metric model and the Klein model of measurement, Minkowskispace curvature of spacelike surfaces, complex transformations, complex analytic functions,Riemann surfaces, Conformal coordinate.

9 , Beltrami equation, spherical measure and Lobachevsky metric, a space of constant curvature, matrix, matrices, the exponential map of the surface in space.

32 。 Derivative of a vector-valued functions, differentiable vector fields on affine space and its basic properties.

10 , Four-element number, the Conformal metric, Conformal transformations,Liouville theorem, differentiability of a vector field, directional derivative and the covariant derivative and the covariant differential and intrinsic differential and its basic properties.

Liaison, Surface intrinsic differential operators on the coordinates, Christoffel Symbols,

35 。 Christoffel symbol symmetry,Christoffel identity.

Gauss Formulas, Weingarten Equation.

11 , Parallel vector fields, the GEODESIC curvature of the curve on a surface, GEODESIC, and

37 。 GEODESIC equations, existence and uniqueness of geodesics and geodesics on the luobaqiefusijiping.

38 。 Through two points of GEODESIC, GEODESIC RADIUS.

39 。 GEODESIC distance, in Compact sets limits on the relationship.

40 。 Half-GEODESIC coordinates on a two-dimensional surface.

41 。 As shortest geodesics on two-dimensional surfaces.

In parallel moves,

42 。 Offset curves parallel moves, the determinant of the derivative of a vector field.

43 。 Offset curves on surfaces, speed and angle of rotation of a vector field.

44 。 Bounded surfaces parallel to the angle between the vectors and moving, rotation of a vector field and measure the sum of the Interior angles of a triangle.

45 。 Bounded surfaces of vector angles and parallel relationship between curvature and movement.

Shortest path theorem, Gaussian curvature invariant Gauss Great theorem, Gauss Equation, Codazzi-Mainardi Curvature tensor, equation, theorem, the local surface theory Gauss Parallel to the curvature, GEODESIC coordinates.

12 , Surface of isomorphism,Maurer-Cartan equations, GEODESIC curvature,Gauss-Bonnet theorem. The Interior angles of a triangle on a sphere and luobaqiefusijiping.

13 , Widespread nature of surfaces,Riemann and pseudo- Riemann space of tensors and fake a one-parameter group of diffeomorphisms, indices of a vector field maps.

47 。 Surface of sphere mapping.

48 。 Complex structures on the surface, spherical complex structures and complex form of Conformal Euclidean metric.

Differential Geometry

1 , B.A.Dubrovin 、 A.T.Fomenko 、 S.P.Novikov Modern geometry.

2 , P.K.Rachevsky , Differential geometry tutorials 13-58 (In addition to 23 、 29 、 30 、 33 、 43 Section) and 86-88Sections.

3 , S.P.Novikov 、 A.T.Fomenko And preliminary differential geometry and topology, the first part.

4 , A.S.Mishchenko 、 A.T.Fomenko , Differential geometry and topology, a simple tutorial, 1 Chapter 1 、 2 Section, subsection 2 Chapter 4 Section, subsection 4 Chapter, 5 Chapters.

5 , P.K.Rachevsky , Riemannian Geometry, and tensor analysis, 44-48 Sections.

A? C? Feijinke differential geometry problem set, Beijing Normal University Press

In the differential geometry theory and exercises puxici

A.T.Fomenko Differential geometry and topology ,Consultants Bureau
Dubrovin, Fomenko, Novikov “Modern geometry-methods and applications”Vol 1—3
A Comprehensive Introduction to Differential Geometry vol 1-5 ,by Michael Spivak 

Differential Geometry of Curves and Surfaces by Manfredo Do Carmo
M. Postnikov, Linear algebra and differential geometry ,Mir Publishers

W.M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry Academic Press, Inc., Orlando, FL, 1986

Chen Qing and Chia Kuai Peng, Differential Geometry

Eisenhart “Diffenrential Geometry(?)”

N. Hicks, Notes on differential geometry, Van Nostrand.

Hilbert ,foundations of geometry ;

T. Frenkel, Geometry of Physics

Peter Petersen, Riemannian Geometry :

Riemannian Manifolds: An Introduction to Curvature by John M. Lee :

Helgason , Differential Geometry,Lie groups,and symmetric spaces :

Lang, Fundamentals of Differential Geometry :

kobayashi/nomizu, Foundations of Differential Geometry :

Riemannian Geometry I.Chavel :

Darboux “Lecons sur la theorie generale des surfaces” 。

Gauss “Disquisitiones generales circa superficies curvas” 。

P.Dombrowski “150 years after Gauss’ ‘Disquisitiones generales circa superficies curvas’ “

R.Osserman “Lectures of Minimal Surfaces”

J.C.C.Nitsche “Lectures on Minimal Surfaces”(Vol.1)

Shiing-Shen Chern, lectures on differential geometry, North Edition
Chen Weihuan, the differentiable manifold of the preliminary higher education version
Su buqing , Of the differential geometry of the higher education version

Wu Daren ” Differential geometry (?)”, Lectures on differential geometry of the higher education press

Shen, chunli , Huang Xuanguo ” Differential geometry “( Economic science press , 97) 。

Jiang Guoying , Huang Xuanguo ” Differential geometry 100 ” 。

Peng Jia GUI of the differential geometry of the higher education press

Shiing-Shen Chern, lectures on differential geometry of Nankai University

178 Of the differential geometry 4 Mei Xiangming , Huang Jingzhi series

181 Of the differential geometry of the zhoujianwei the

185 Lectures on differential geometry of the Wu

“The problem sets and books”

187 Yang wenmao the differential geometry problem set , Fu Chaojin , Written by Chen xinyue

188 Puxici the differential geometry theory and exercises

189 Of the solutions of differential geometry study guide and exercises selected Mei Xiangming, Wang Huichun series

“Increase”

191 The differential geometry of five talk with Su buchin

192 Lectures on differential geometry, Shing-Tung Yau, Sun Licha on

193 Liang Canbin the introduction to differential geometry and general relativity, now the

Differential Forms

X

(Geometry of)Manifolds

Lang, Differential and Riemannian manifolds :

Warner,Foundations of Differentiable manifolds and Lie groups :

Introduction to Smooth Manifolds by John M. Lee :

1.W.M.Boothby “An Introduction to Differentiable Manifolds and Riemannian Geometry”

B.A. Dubrovin, A.T. Fomenko, S.P. Novikov “Modern Geometry–Methods and Applications” The first , Two volumes

Gallot, Hulin, Lafontain “Introduction to Riemannian Geometry”(?)

J. Milnor differential Topology from a point of view ( translation : from the viewpoint of differential topology)

J. Milnor Morse Theory ( translation : Morse theory ) About

Spivak “Calculus on Manifolds”(?) ( Chinese name called ” calculus on manifolds”).

V.I.Arnold “Mathematical Mathods of Classical Mechanics”

R.Narasimhan “Analysis on Real and Complex Manifolds”

C. von Westenholz “Differential forms in Mthematical Physics”

Shiing-Shen Chern , Chen Weihuan ” Preliminary differential geometry”

Zhengguo Bai , Shen yibing , Shui Naixiang , Guo Xiaoying ” Initial Riemannian Geometry ” 。

Su Jingcun ” The topology of manifolds “. This book was a large , The informative , And there are many authors and , Interesting . There is a book , May not import expert discernment , But I think it is very good,

Bott, Raoul, R. Bott, and Loring W. Tu. Differential Forms in Algebraic Topology (Graduate Texts in Mathematics; 82). Reprint edition. New York: Springer-Verlag, June 1, 1995. ISBN: 0387906134.

Buy at Amazon Abraham, Ralph, Jerrold E. Marsden, and Tudor Ratiu. “Manifolds, Tensor Analysis, and Applications.” Applied Mathematical Sciences. Vol. 75. New York: Springer Verlag, May 1, 1998. ISBN: 3540967907.

Buy at Amazon Hirsch, Morris W. “Differential Topology.” Graduate Texts in Mathematics. Vol. 33. Reprint edition. New York: Springer-Verlag, November 1, 1988. ISBN: 0387901485.

Geometric Analysis

Peter Li

Yau

Differential geometry and topology

4 。 Tangent vector with hypersurface tangent, curvature tensor of differentiable functions.

5 。 Smooth maps on differentiable manifolds.

1 , Vector field, the integral curves of the vector field and his smooth, flow, tubular neighborhood,

7 。 Manifold tangent vector bundle, smooth vector bundles.

8 。 Regular map dirty form inverse mapping of the structure.

9 。 Canonical mapping is a compact inverse mapping of vector bundles on a smooth manifold.

10 。 The commutator of vector fields, and its nature.

11 。 Holomorphic vector bundle its properties.

Fiber bundles and vector bundles, the sphere bundle, topological groups, track space.

2 , Lens spaces, homotopy, homology of map, homotopy, fundamental group, the basic group of operations, path lifting lemma, homotopy lifting lemma, the orbit space of fundamental groups, basic groups of product space.

3 , Homotopy types, deformation retraction, contractible space, theBrouwer fixed point theorem, theJordan curve theorem, surface boundaries, simplex, simple partitioning, simplicial complex, but simple partitioning space, Barycentric subdivision.

4 , Hosting, simplicial approximation theorem, complex edge group, andVan Kampen theorem, the orbit space of simple triangulation, infinitely complex.

5 , Classification of closed surfaces, surfaces of orientability, andEuler characteristic number, the symbols, the genus of the surface.

1 。 Limited overwrite, decomposition theorem.

2 。 Compact manifolds embedded in Euclid space.

12 。 Linear differential form, some basic linear differential form.

13 。 Tensor fields, local properties.

14 。 Tensor addition, multilinear function, tensor, tensor fields.

15 。 Contraction operation, for example.

16 。 Symmetric and skew-symmetric tensor, staggered symmetric operators.

19 。 Basic information of the skew-symmetric tensor space, differential forms the base and coordinate.

20 。 Differential form under the coordinate transformations and coordinate transformations of base maps.

21 。 To the manifold, to books, to differential forms.

22 。 Exterior derivative, exterior differential operation that coordinates.

23 。 Three dimensional space vector field of differential operation.

24 。 De Rham cohomology,de Rham theorem.

25 。 De Rham cohomology of differential forms on a manifold of smooth maps.

26 。 De Rham cohomology of homotopy properties.

27 。 De Rham cohomology of the homotopy invariance ofPoincare theorem.

6 , Riemann metric,Riemann manifolds, and Riemann metric of a smooth manifold. Riemann product manifold,Riemann manifolds, andRiemann immersion, complex projective space, homogeneous Riemann space, Steenrod Theorem, contact, Levi-CivitaLiaison, Riemann Submanifolds in contact.

28 。 Manifold and the integration of differential forms on a manifold.

29 。 Riemann integral of a function on a manifold and its comparison with the differential form of integral, and three dimensional space curved surface and curve integral of a vector field on.

30 。 Stokes formula, special forms and results (three dimensional space of vector fields and thede Rham cohomology).

31 。 Euclid space derivative of a vector field, and its basic properties.

32 。 Affine contact absolute derivative of a vector field, coordinate,Christoffel symbols, equivalent contact contact the affine equivalence.

33 。 Symmetric affine connection.

34 。 Vector differential and moving parallel to the vertical curve, tangent space transformation properties of differential operators on a vector field.

35 。 Curve moves along the vectors with arbitrary tensors, tensor products and curl.

36 。 Tensor’s covariant derivative.

37 。 Number of vector and tensor coordinates of differential forms, parallel condition, the gradient of a tensor, vector field of fork.

38 。 Riemann coordination condition of the affine connection on a manifold.

39 。 Riemann affine connection on a manifold and uniqueness of the Levi-Civita theorem.

40 。 Euclidean space, pseudo-Euclidean space and Riemann manifold of submanifolds of covariant derivative and affine connection.

41 。 And its equation of GEODESIC, GEODESIC equations of existence, specify the direction of a point on the existence and uniqueness of geodesics.

42 。 Riemann curvature of manifolds andRiemann manifold of the GEODESIC curvature of the curve on the manifold, ball and Lobachevsky plane of curvature.

7 , Covariant derivative along a curve, parallel move, GEODESIC, GEODESIC local existence and uniqueness, exponential map,Gauss ‘s lemma, complete Riemann manifolds and theHopf-Rinow theorem.

43 。 Elementary calculus: extreme value of Lagrange and Euler equations.

44 。 As action and extreme value for the functional length of the GEODESIC.

45 。 GEODESIC nature of the neighborhood, and extensibility, GEODESIC completeness of geodesics on a manifold of infinite extension.

46 。 Connectivity on a closed set and geodesics, normal coordinates, measuring the Earth’s surface, measuring the radius of the Earth’s surface, their orthogonality and Geodesy extrema.

47 。 Riemann curvature tensor of the manifold.

48 。 Properties of curvature and its coordinates.

49 。 Riemann metric symbols.

50 。 Two-dimensional Riemann curvature of the manifold, as a locally Euclidean space of zero curvature.

51 。 Sectional curvature, as measured the total curvature of a surface.

53 。 Two-dimensional Riemann manifolds the rotation tensor field along the closed road.

54 。 Around the closed curve on a two-dimensional manifold a vector measure the sum of the Interior angles of a triangle.

55 。 Geometric mean curvature tensor (a closed curve surrounding the relationship between the vector and bivector).

56 。 Two-dimensional geometric curvature of the manifold along a given direction.

8 , Cut locus, the second covariant derivative, the curvature tensor algebra properties, calculation of curvature,Ricci curvature, the scalar curvature, the second variation of the first variational forms, forms,Jacobifield, level.

9 , O ‘ Neill formula, formal homogeneity metric,Gauss ‘s lemma, conjugate, with constant sectional curvature of space and theMyers theorem,Hadamard theorem, a differentiable manifold on the measurable sets, Volume estimates, exponential growth of finite groups, andMilnor-Wolf theorem.

10 , Fundamental groups of compact manifolds of negative curvature growth, andMilnor theorem,Gauss-Bonnetformula,Gromov theorem, theCheeger theorem, Conformally flat manifolds, the second Bianchi identity, simple

Homology groups, edge closed, oriented simplexes, homology group, homology classes, simple mapping, chain complex, Web type.

11 , Maps, continuous vector field,Euler-Poincare formula, rational coefficient homology group,Borsuk-Ulamtheorem,Lusternik theorem,Lefschetz Fixed point theorem andHopf theorem.

12 , Dimension and knot equivalence, knot groups, andSeifert surface, covering spaces, mapping, lifting theorem, the universal covering space.

13 , Chain ring,Kauffman knot polynomial,Jones knot polynomial,Conway knot polynomial,Alexander knot polynomial, Vassiliev knot invariants, andKontsevich theorem.

1 , S.P.Novikov 、 A.T.Fomenko And preliminary differential geometry and topology, science press, 1987 。

2 , B.A.Dubrovin 、 S.P.Novikov 、 A.T.Fomenko Modern geometry, science press, 1985 。

3 , P.K.Rachevsky , Riemannian Geometry, and tensor analysis, technology and literature Publishing House, 1953 。

4 , A.S.Mishchenko 、 A.T.Fomenko , Course in differential geometry and topology, faketeliya Publishing House, 2000 。

5 , S.P.Novikov 、 I.A.Taimanov Modern geometry and field theory, independent of Moscow University Press, 2004 。

6 , J.Milnor , Morse Theory of Moscow book publishers in the world, 1985 。

Chen Weihuan, an introduction to Riemannian Geometry (upper and lower), Peking University
Wu Hongxi, of the initial Riemannian Geometry, North Edition

Application of geometric problems

1 。 Configuration space, examples of non-trivial structure space, planar pendulum with three-dimensional, and secondary compound pendulum movement of the rigid body around a fixed point.

2 。 Phase space, for example.

3 。 Manifold.

4 。 Variational problems, Euler’s equation.

5 。 Symplectic manifolds and symplectic space is defined.

6 。 The symplectic form on the symplectic space and its nature.

7 。 Hamilton functionHamilton equations.

8 。 Oblique derivative function,Poisson brackets.

9 。 Score for the first time.

10 。 Vector fields and their distribution.

11 。 Frobenius theorem of complex forms.

12 。 Symplectic form into standard form,Darboux theorem.

13 。 Exchange and the Diffeomorphism Group of a vector field.

14 。 Symplectic manifolds on the Diffeomorphism Group of finite algebras.

15 。 And symmetric integrals for the first time the Netter theorem.

16 。 Completely integrable Hamilton System Liouville theorem.

17 。 Non-commutative situation completely Integrable systems.

18 。 The integrability of the rigid body dynamics.

19 。 The concept of differential operators on manifolds.

20 。 The pseudo differential operators on manifolds.

21 。 Sobolev space of pseudodifferential operators with the Sobolev norms.

22 。 On the Sobolev space of compactness of Sobolev theorem.

23 。 Fredholm operators and compact operators.

24 。 Fredholm operators index and its properties.

25 。 Fredholm alternative theorem.

26 。 Vector bundles and elliptic operator.

27 。 Atiyah–Singer index theorem.

1 , M.Hirsh , Differential Topology , Springer , 1976 。

2 , R.Thom And some general properties of differentiable manifolds, loading ” Fiber spaces and its applications ” A book, Moscow, foreign languages Publishing House, 1958 。

3 , V.V.Trofimov 、 A.T.Fomenko , Differential equation of integrable Hamiltonian of algebra and geometry, faketeliya Publishing House, 1995 。

4 , V.I.Arnold 、 V.V.Kozlov 、 A.I.Neyshtadt , Mathematical topics in classical mechanics and celestial mechanics, science and technology information Research Institute of the Soviet Union, 1985 。

5 , A.S.Mishchenko Fiber bundle and its application, science press, 1984 。