教程的内容主要包括：

1. reviews on fundamentals of riemannian geometry;

2. Jacobi fields and applications

3. space forms

4. second variation of arc length (or energy) functional and applications

5. Morse theory and applications

6. comparison theorems and applications

7. selected topics on curvature and topology

主要参考文献：

1. 杨义虎：黎曼几何讲义（incomplete）

2. Manfredo do Carmo: Riemannian geometry

3. J. Cheeger & D. Ebin, Compartison theorems in Riemannian geometry

要求：完成do Carmo书中相应章节的的习题

12. 微分几何（2021秋季，本科生课程；时间：周一10：00-11：40 (地点：下院509)，周三10：00-11；40（地点：下院410）

第一章、空间曲线（局部理论）

1. 正则参数曲线、弧长参数

2、空间曲线的曲率、挠率和Frenet标架及Frenet公式

3、空间曲线的标准形式

4、空间曲线的基本定理

5、平面曲线

6、习题

第二章、空间曲面（局部理论）

1、正则参数曲面（切平面，映照（函数）的微分），第一基本形式

2、曲面的Gauss映照及其性质，第二基本形式（法曲率，主曲率，Gauss曲率，平均曲率；及它们在局部坐标下的表示公式）

3、曲面上的特殊曲线（曲率线，渐进曲线，测地线）

4、一些特殊曲面（旋转曲面，函数的图，直纹面，极小曲面）

5、向量场，特殊坐标的构造

6、Gauss定理（曲面的运动方程，结构方程）

7、曲面的基本定理

8、习题

第三章、空间曲面的内蕴几何

1、正则曲面，等距，共性映照

2、共变导数，测地线

3、指数映照，完备性，测地凸邻域

4、Gauss-Bonnet公式

5、习题

第四章、整体微分几何选讲

参考文献

1. 彭家贵、陈卿，微分几何，高等教育出本版社

2. Manfredo do Carmo, Differential Geometry of curves and surfaces

3. W. Klingenberg, A Course in Differential Geometry

11. 微分流形与微分几何（2021秋季，研究生学位课程；时间：周二10：00-11：40，周四10：00-11；40，地点：理科群楼6号楼901）

第一章、微分流形

1、微分流形（局部坐标，微分结构，例子）

2、切空间和余切空间（可微函数、映照及它们的微分）

3、子流形（浸入，嵌入）

4、切向量场（李括号，单参数变换群，李导数）

5、分布与FROBENIUS定理（积分流形与偏微分方程）

6、Morse-Sard定理

7、李群引论

8、习题

第二章、外微分和流形上的积分

第三章、向量丛和联络

第四章、黎曼几何初步

10. Differential Geometry II(Spring of 2020, for undergraduates and graduates; Times: (周三, 11-13节, 闵行 东上院101):

Part I:Some classical topics in the global aspects of curves and surfaces in Euclidean 3-space

Lecture 1. review on (local) differential geometry of curves and surfaces; isometry; completeness of regular surfaces (or abstract surfaces with Riemannian metrics) and Hopf-Rinow theorem

Lecture 2. rigidity of sphere

Lecture 3. Hopf theorem on immersed sphere of constant mean curvature

Lecture 4. Alexandrov theorem and a brief introduction to Hopf conjecture on immersed compact surfaces of cmc (e.g. and Wente‘s examples)

Lecture 5. Hilbert theorem

Lecture 6. Schwarz lemma and its generalizations (Ahlfors-Schwarz lemma.....)

Part II:Vector bundles and connections, Riemannian manifolds

1. review of manifolds (including differential forms and exterior differentiation, vector fields and Lie bracket, and Frobenius Theorem）

2. vector bundles

3. connections on vector bundles

4. curvature tensor and *a brief introduction to Chern classes

4*. geometric significance of curvature; remarks on flat vector bundles

5. fundamental theorem of Riemannian geometry and definitions of various curvatures

6. geodesics, exponential map, and completeness of Riemannian manifolds

..........

References

[1]J. Cheeger & D. Ebin, Comparison theorems in Riemannian Geometry, AMS Chelsea Publishing, 1975

[2] M. do Carmo,Differential Geometry of Curves and Surfaces, Dover Publications, INC., revised & updated second edition, 2016:Chapter 5.

[3] 龚昇， 简明复分析， 北京大学出版社.

[4] H. Hopf,Differential Geometry in the Large, Springer-Verlag, 1983:Part II.

[5] J. Jost,Riemannian Geometry and Geometric Analysis, Springer, Seocnd ed., 1998:Chapter 3, 4.

[6] R. Osserman, A Survey of Minimal Surfaces, Dover, 2014.

[7] P. Petersen,Riemannian Geometry, Springer, Second edition.

[8] 伍鸿熙,黎曼几何初步, 北京大学出版社, 1989.

[9] Weiping Zhang: Lectures on Chern-Weil theory and Witten deformations, Nankai Tracts in Mathematics, Vol.4.

9. Calculus on manifolds(Summer school of 2019, for undergraduates; Tue, Fri. 2:00-5:40,19-22周,下院310)

We‘ll (strictly) use Manfredo P. do Carmo‘s book:Differential Forms and application, Universitext, Springer.

We expect to finish at least Chapters 1, 3, 4, so that in the next step for study of geometry, we can start with some more advanced materials, like Riemannian and complex geometry, the geometry of (real or complex) vector bundles, and some fundamental tools of geometric analysis.

If time admitted, we also hope talk about Chapters 5, 6, which is actually part of elementary differential geometry, but treated by using differential forms.

Wish that the students can finish all exercises in the book; this is a necessary step to understand the materials in the book and (be able to) use them in the future. Essentially, this course provides only with a suitable language for modern mathematics, in particular geometry.

Prerequisites: Some calculus of several variables and linear algebra, and some point-set topology.

8. Differential Geometry II(Spring of 2019, for undergraduates and graduates; Times: Tue. 6-8节; Place: 下院401):

Part I:Some classical topics in the global aspects of curves and surfaces in Euclidean 3-space

Lecture 1. review on (local) differential geometry of curves and surfaces; isometry; completeness of regular surfaces (or abstract surfaces with Riemannian metrics) and Hopf-Rinow theorem

Lecture 2. rigidity of sphere

Lecture 3. Hilbert theorem

Lecture 4. Hopf theorem on immersed sphere of constant mean curvature and a brief introduction to imbedded or immersed compact surfaces of cmc (e.g. Alexandrov theorem and Wente‘s examples)

Part II:Vector bundles and connections

1. review of manifolds (including differential forms and exterior differentiation, vector fields and Lie bracket, and Frobenius Theorem）

2. vector bundles

3. connections on vector bundles

4. curvature tensor and a brief introduction to Chern classes

4*. geometric significance of curvature; remarks on flat vector bundles

5. fundamental theorem of Riemannian geometry and definitions of various curvatures

6. geodesics, exponential map, and completeness of Riemannian manifolds

Part III:Introduction to complex geometry

1. complex manifolds and complex structures, almost complex structure and Newlander-Nirenberg Theorem

2. holomorphic vector bundles, sheaves and sheaf-cohomology, exponential sheaf sequence

References

[1] M. do Carmo: Differential Geometry of Curves & Surfaces, revised & updated 2nd edition, Dover;

[2] S. Donandson: Riemann surfaces, Oxford Graduate Texts in Math., 22, Oxford University Press;

[3] P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley Classics Library;

[4] S. Kobayashi and K. Nomizu: Foundations of Differential Geometry, Vol. I, Wiley Classics Library;

[5] N. Mok, Metric Rigidity Theorems on Hermitian Locally Symmetric Manifolds, Series in Pure Mathematics, Vol. 6, World Scientific;

[6] C. L. Siegel: Topics in Complex Function Theory, Vol. 1: Elliptic Functions and Uniformization theory, Wiley Classics Library;

[7] Weiping Zhang: Lectures on Chern-Weil theory and Witten deformations, Nankai Tracts in Mathematics, Vol.4.

7. Differential Geometry(Fall of 2018, for undergraduates, the following [1, 2] will be the main references; Times: Tue. and Thur., 10:00-11:40; Place: Middle Teaching Building 212):

This is a course for undergraduates. It mainly concerns the geometry of curves and surfaces in Euclidean spaces, especially 3-space.We mainly concern local aspects but also some global aspects of curves and surfaces.

To study the global aspects of surfaces and also let students understand parameters being artificial and accept the invariant view-point as early as possible, we‘ll try to start with the notions of regular non-parametrized (or abstract) surfaces (similarly for curves)---2-dim manifoldsand 2-dim Riemannian manifolds.

Also, we‘ll introduce some general notions of Riemannian geometry (but restricted to the 2-dim case):connection, geodesic, exponential map, completeness, etc. Futhermore, we‘ll informally introduce topological classification of closed orientable surfaces by nonnegative integers---genus---by means of tirangulation. Then, we‘ll prove the famous Gauss-Bonnet formulae.

Prerequisite: Calculus, Linear algebra, Analytic geometry, some point-set topology (of Euclidean space)

Note: the materials with asterisk areNOTin the teaching plan.

Chapter 1Curves in Euclidean 3-spaces--local theory

1. regular (parametrized and non-parametrized) curves, arc length parameter; tangent vector, normal and binormal vectors, osculating plane, normal plane and rectifying plane

2. Frenet frame and Frenet formulae, curvature and torsion; canonical (normal) form near a point of curves; geometric implication of curvature and torsion; plane curves

3. fundamental theorem for curves in 3-space (uniqueness and existence to a curve with arc length parameter in 3-space with prescribed curvature (>0) and torsion)

Exercises:1. compute the curvature and torsion of a curve under general regular parameters;

2. think why "curvature" and "torsion" are (geometric) invariants of a space curve---independent of choice of parameters;

3. derive the canonical (normal) form at a point of a 3-space curve and show the geometric meaning of curvature and torsion;

4. use the normal form of a curve to understand Corollary 1.5.4 and draw the projections in the corresponding planes;

5. finish Ex. 1.6.4.

*Some additional readingsfor Chap. 1 (some global aspects of plane curves):

1. Chap. 2 of the textbook;

2. (general) 4 vertex theorem and its converse ([1] D. DeTurck, H. Gluck, D. Pomerleano, and D. Shea Vick, The four vertex theorem and its converse, Notices of AMS, Vol. 54, No. 2, 192-207; [2] Bjoern E. J. Dahlberg, The converse of the four vertex theorem, Proc. AMS, Vol. 133, No. 7, 2131-2135) .

Chapter 2Surfaces in Euclidean 3-spaces--local theory

1. regular (parametrized and non-parametrized) surfaces: tangent space and tangent vectors (fields), changes of variables of surfaces, differentiable functions and (the differential or tangent map of) a differentiable map between surfaces; vector fields along surfaces: tangential (normal) vector fields, coordinate vector fields; unit normal vector field of surface---Gauss map; orientable surfaces

2.the 1st fundamental form: independent of parameters (so a geometric invariant of the corresponding non-parametrized surface)；area, angle and length of curves on surfaces; isometries of surfaces, examples

The following 3 sections can be considered as the geometry of the Gauss map

3.the 2nd fundamental form: independent of parameters (so a geometric invariant of the corresponding non-parametrized surface); examples；curves on surfaces: line element；normal curvature, Meusnier‘s theorem, asymptotic directions (curves)；Weingarten map: principal directions (curvature), curvature lines Roderiques theorem, Gauss curvature, mean curvature；canonical form of a surface at a point: elliptic, parabolic, and hyperbolic points

4. vector fields and their trajectories and first integrals；coordinate system generated by two vector fields which are linearly independent at some points: orthogonal coordinate systems; equation of asymptotic curves, coordinate system of asymptotic curves；equation of curvature line, principal curvature coordinate system (coordinate system of curvature lines)

5. Gauss map and geometric explanation of Gauss curvature: geometry of second fundamental form is equivalent to geometry of Gauss map; minimal surfaces: critical points of area functional

6. some special surfaces: ruled surfaces and developable surfaces, classification of developable surfaces; surfaces of revolution with constant Gauss curvature (pseudo-sphere) and minimal (zero mean curvature) surfaces: catenary, catenoid, etc

Exercises: 1. Ex.4,7,8 of Section 2-5 in [1];

**Try to prove: a regular compact surface without boundary in 3-Euclidean space is orientable.

2. Prove the remark in Page 45 in [4]

3. Prove 3.9.1, 3.9.2, 3.9.3, 3.9.4, 3.9.6, 3.9.7, 3.9.8*(5.7.4);

4. write the Gauss‘ equation under orthogonal coordinates;

5. write Mainardi-Codazzi equations under principal directions coordinate systems (parameter net of lines of curvature)

Chap. 3Intrinsic geometry of surfaces in Euclidean 3-space

1. equations of motion for surfaces and structure equations (compatibility equations): Gauss‘s theorema egregium; fundamental theorem for surfaces in Euclidean 3-space

2. covariant differentiation (of vector fields); parallel translation; geodesic curvature (and relation to normal curvature), Liouville formula; geodesics and its equations

3. (local) Gauss-Bonnet theorem for simple closed domains with piece-wise smooth boundary in a surface

4. exponential map, geodesic polar coordinate, Gauss lemma, (local) minimality of geodesics; surfaces of constant curvature

5. intrinsic generalization of regular (non-parametrized) surfaces in Euclidean 3-space: 2-dimensional (abstract) manifolds and 2-dimensional (oriented) Riemannian manifolds, isometries (and conformal mappings); tangent spaces and tangent vectors (fields); Riemannian covariant differentiation (-Levi-Civita connection); Lie bracket of smooth vector fields, curvature tensor, curvature, geodesics, completeness

6. (global) Gauss-Bonnet theorem for compact surfaces with or without boundary: triangulation of surfaces, Euler characteristic; topological classification of (oriented) closed surfaces; Gauss-Bonnet theorem

Exercises: 1. Ex.5, 6, 8* of Section 4-4 in [1];

2. show that any geodesic of the revolution paraboloid (旋转抛物面) z=x^2+y^2, if it is not a meridian (子午线)，intersects itself an infinite number of times; (I hope, not only a good understanding in geometry, but also a good writing)

3. (using Liouville formula or the equations of geodesic) find the equation of the following geodesic c of the (abstract) surface S: Let S be the upper half plane with the coordinate (u, v) and the 1st fundamental form (Riemannian metric) g=v(du^2+dv^2), the geodesic c through the point (0, 1) and having an angle \theta_0 with the u-direction.

4. Ex. 1, 2, 3, 7, 8, 9 of Section 4-6 in [1]

[1] Manfredo do Carmo:Differential Geometry of Curves & Surfaces, revised & updated 2nd edition, Dover

[2] 彭家贵，陈卿:微分几何， 高等教育出版社

[3] J. Simons,Minimal varieties in Riemannian manifolds, Ann. Math., 88(1968), 62-105

[4] W. Klingenberg:A courses in Differential Geometry, Springer-Verlag

6. Differential Geometry, II(spring of 2018, for undergraduates and graduates; Monday 12:55-15:40)

This is a continuation of a course of Differential Geometry in Fall of 2017. This course is split into two parts. The first part is concerning the global aspects of curves and surfaces in Euclidean 3-space; the second one is concerning some rudiments of Riemann surfaces and Kaehler geometry..........

Part I:we‘ll lecture some classical topics in the global aspects of curves and surfaces in Euclidean 3-space, like isoparametric inequality, four vortex theorem, rigidity of the sphere, Hilbert theorem, and Hopf and Alexsandrov theorems on constant mean curvature; we also wish to give a brief introduction of minimal surfaces and their Gauss maps in Euclidean 3-space, especially Bernstein theorem.

Lecture 1. review on (local) differential geometry of curves and surfaces, (nonparametrized) regular surfaces, differentiable maps and their differentials, immersion and imbedding, (local) isometry; *completeness of (nonparametrized) regular surfaces and abstract surfaces with metrics and Hopf-Rinow theorem

Lecture 2. rigidity of sphere

Lecture 3. Hilbert theorem (Finish do Carmo‘s book 5.11: Ex. 1, 2)

Lecture 4. Fenchel and Fary-Milnor theorems on regular, simple and closed curves

Lecture 5. Hopf theorem on immersed sphere of constant mean curvature and *a brief introduction to imbedded or immersed compact surfaces of cmc (e.g. Alexandrov theorem and Wente‘s examples)

Part II:will concern some basics of Riemann surfaces and Kaehlerian manifolds.

Lecture 1. review of differentiable manifolds: definition, (co)tangent spaces, (co)tangent bundles, vector bundles, tensor product, sections (e.g. differential forms, vector fields), Lie bracket of vector fields, exterior differential operator; submanifolds, Frobenius theorem; connections on vector bundles and curvature tensors

Lecture 2. Riemannian manifolds and submanifolds: Riemannian metrics and connection, Riemannian curvature tensor, sectional, Ricci, and scalar curvature; a brief introduction to curvature and topology; Riemannian submanifolds, induced connections, second fundamental form and mean curvature (vector), Gauss equation, Codazzi equation and Ricci identity

Lecture 3. complex manifolds and complex structure, holomorphic (co)tangent bundle, Hermitian and Kaehler metrics, connections (Riemannian and Hermitian connections and their consistency under the Kaehlerian condition), holomorphic (bi-)sectional curvature

5. Differential Geometry(Fall of 2017, for undergraduates, the following [1] will be the textbook; Tuesday and Thursday, 10:00-11:40, Eastern Top Teaching Building 212):

This is a course for undergraduates. It mainly concerns the geometry of curves and surfaces in Euclidean spaces, especially 3-space.We mainly concern local aspects but also some global aspects of surfaces.

To study the global aspects of surfaces, we‘ll try to introduce the notions of abstract surfaces---2-dim manifoldsand 2-riemannian manifolds;and in turn we‘ll introduce some general notions of riemannian geometry (but restricted to the 2-dim case):geodesic, exponential map, completeness, Jacobi fields, conjugate points, and comparison theorems (if time admitted) etc. Futhermore, we‘ll informally introduce topological classification of closed orientable surfaces by nonnegative integers---genus---by means of tirangulation. Then, we‘ll prove the famous Gauss-Bonnet formulae.

Some preliminaries:topology of Euclidean space; tangent space and tangent bundle, differential (tangent map) of a differentiable map; local behaviour of differentiable map (inverse function and implicit function theorems)

Note: the materials with asterisk areNOTin the teaching plan.

Chapter 1Curves in Euclidean 3-spaces

1. regular (parametrized) curves, arc length parameter; tangent vector, normal and binormal vectors, osculating plane, normal plane and rectifying plane

2. Frenet frame and Frenet formulae, curvature and torsion; canonical (normal) form near a point of curves; geometric implication of curvature and torsion; plane curves

3. fundamental theorem for curves in 3-space (uniqueness and existence to a curve with arc length parameter in 3-space with prescribed curvature (>0) and torsion)

Exercises:1. compute the curvature and torsion of a curve under general regular parameters;

2. think why "curvature" and "torsion" are (geometric) invariants of a space curve---independent of choice of parameters;

3. derive the canonical (normal) form at a point of a 3-space curve and show the geometric meaning of curvature and torsion;

4. use the normal form of a curve to understand Corollary 1.5.4 and draw the projections in the corresponding planes;

5. finish Ex. 1.6.4.

*Some additional readingsfor Chap. 1 (some global aspects of plane curves):

1. Chap. 2 of the textbook;

2. (general) 4 vertex theorem and its converse ([1] D. DeTurck, H. Gluck, D. Pomerleano, and D. Shea Vick, The four vertex theorem and its converse, Notices of AMS, Vol. 54, No. 2, 192-207; [2] Bjoern E. J. Dahlberg, The converse of the four vertex theorem, Proc. AMS, Vol. 133, No. 7, 2131-2135) .

Chapter 2Regular (parametrized) surfaces in Euclidean 3-spaces--local theory

1. regular (parametrized) surfaces, tangent space (tangent vectors), changes of variables of surfaces, unparametrized surfaces; vector fields along surfaces: tangential (normal) vector fields, coordinate vector fields; unit normal vector field of surface---Gauss map, Gauss frame;differentials of composed maps (the special case of changes of variables)

2.the 1st fundamental form: independent of parameters (so it is the geometric invariant of the corresponding unparametrized surface)

3.the 2nd fundamental form: independent of parameters (so it is the geometric invariant of the corresponding unparametrized surface);Weingarten map(transformation) of tangent spaces of surfaces; examples.

4. curves on surfaces: line element, Meusnier‘s theorem, normal curvature

5. Weingarten map: principal curvature, Gauss curvature, mean curvature

6. canonical form of a surface at a point: elliptic, parabolic, and hyperbolic points; vector field and its trajectories and first integral; coordinate system generated by two vector fields which are linearly independent at some points: orthogonal coordinate systems; principal directions and (equation of) lines of curvature, Rodriques‘ Theorem, principal curvature coordinate system (coordinate system of curvature lines); asymptotic directions and (equation of) asymptotic curves, coordinate system of asymptotic curves

7. ruled surfaces and developable surfaces: classification of developable surfaces

8. Gauss map and geometric explanation of Gauss curvature: geometry of second fundamental form is equivalent to geometry of Gauss map; minimal surfaces: critical points of area functional

9. surfaces of revolution with constant Gauss curvature (pseudo-sphere) and zero mean curvature (catenary and catenoid) (Ex.)

Exercises: 1. Ex.4,7,8 of Section 2-5 in [2];

2. Prove the remark in Page 45;

3. Prove 3.9.1, 3.9.2, 3.9.3, 3.9.4, 3.9.6, 3.9.7, 3.9.8*(5.7.4);

4. write the Gauss‘ equation under orthogonal coordinates;

5. write Mainardi-Codazzi equations under principal directions coordinate systems (parameter net of lines of curvature)

Chap. 3Intrinsic geometry of surfaces in Euclidean 3-space--local theory

1. equations of motion for surfaces and structure equations (compatibility equations): Gauss‘s theorema egregium; fundamental theorem for surfaces in Euclidean 3-space

2. vector fields and covariant differentiation; parallel translation

3. geodesic curvature (and relation to normal curvature), Liouville formula; geodesics and its equations

4. (local) Gauss-Bonnet theorem for simple closed domains with piece-wise smooth boundary in a surface

5. exponential map, geodesic polar coordinate, Gauss lemma, (local) minimality of geodesics; isometries, surfaces of constant curvature

Exercises: 1. Ex.5, 6, 8* of Section 4-4 in [2];

2. show that any geodesic of the revolution paraboloid (旋转抛物面) z=x^2+y^2, if it is not a meridian (子午线)，intersects itself an infinite number of times; (I hope, not only a good understanding in geometry, but also a good writing)

3. (using Liouville formula or the equations of geodesic) find the equation of the following geodesic c of the (abstract) surface S: Let S be the upper half plane with the coordinate (u, v) and the 1st fundamental form (Riemannian metric) g=v(du^2+dv^2), the geodesic c through the point (0, 1) and having an angle \theta_0 with the u-direction.

4. Ex. 1, 2, 3, 7, 8, 9 of Section 4-6 in [2]

Chapter 4Selected topics on the global aspects ofintrinsic geometry of surfaces in Euclidean 3-space
(or 2-dimensional Riemannian geometry)

1. regular (non-parametrized) surfaces in Euclidean 3-space; 2-dimensional (abstract) surfaces and tangent spaces; orientability (of surfaces) and 2-dimensional (oriented) Riemannian manifolds; (Riemannian) covariant differentiation (-Levi-Civita connection); Lie bracket of smooth vector firlds, curvature tensor, curvature

2. (global) Gauss-Bonnet theorem for compact surfaces with or without boundary: triangulation of surfaces, Euler characteristic; classification of (oriented) closed surfaces

3. geodesics (exponential map, geodesic polar coordinate, Gauss lemma, (local) minimality of geodesics, as in Section 4 of Chap. 3); (metric, geodesic) completeness, Hopf-Rinow theorem

*4. Jacobi fields and conjugate points

*5. a brief introduction to curvature and geometry and topology of (complete) surfaces: first and second variations of arc length, Bonnet‘s theorem; Cartan-Hadamard theorem

Exercises: 1. Let X, Y, Z, W be smooth vector fields, R the curvature tensor. Prove that 1) the value of R(X, Y)Z) at p depends only on the values of X, Y, Z at p; 2) R(X, Y)Z=-R(Y, X)Z; 3) R(X, Y)Z+R(Y, Z)X+R(Z, X)Y=0; 4) =

2. understand Poincare theorem using Gauss-Bonnet theorem: The sum of the indices of a differentiable vector field with isolated singular points on a closed oriented (without boundary) surface is equal to the Eular Characteristic of the surface. (refer to Section 4.5 of [2]) (it is better if you write a report by yourself after understanding)

*Chapter 5Regular (non-parametrized) surfaces in Euclidean 3-spaces--global theory

In a certain more general framework, the contents of this chapter actually are to concern the (global) aspects of certain special surfaces in Euclidean 3-space, e.g. with certain constraints of Gauss curvature, mean curvature, or topology, etc.. In turn, one can consider submanifolds in a more general ambient Riemannian space and related theory. These are traditional topics in the area of Differential Geometry (see [4]).

Concretely, the topics will includ isoperimetric inequality and Four-vertex theorem in the plane; rigidity of the sphere, Hilbert theorem, Hopf and Alexandrov theorems of closed surfaces with constant mean curvature, and minimal surfaces and Bernstein theorem, etc.. We will talk about these in the course of Differential Geometry II.

References

[1] W. Klingenberg:A courses in Differential Geometry, Springer-Verlag

[2] Manfredo do Carmo:Differential Geometry of Curves & Surfaces, revised & updated 2nd edition, Dover

[3] 彭家贵，陈卿:微分几何， 高等教育出版社

[4] J. Simons,Minimal varieties in Riemannian manifolds, Ann. Math., 88(1968), 62-105

4. A brief introduction to the theory of submanifolds(a mini course for graduate students in geometry)

1) Some preliminaries of vector bundles and connections

2) Second fundamental form of submanifolds： Gauss equation, Mainardi-Codazzi equation, Ricci equation (higher co-dimension)

3) Mean curvature (vector), minimal submanifolds, the first and second variational formulae for volume functional, (semi) stable minimal submanifolds

4) works of Fisher-Colbrie and Schoen, Schoen and Yau about stable minimal 2-dim submanifolds in 3-manifolds with nonnegative scalar curvature

References

[1] H. Blaine Lawson, Jr., Minimal Varieties in Real and Complex Geometry

[2] J. Simons, Minimal varieties in Riemannian manifolds, Ann. Math., 88(1968), 62-105

[3] Y. L. Xin, Minimal Submanifolds and Related Topics

3. Differential Geometry (II)

(Spring of 2017, for undergraduate and graduate; Wednesday, 18:00-20:00, Friday, 18:00-20:00; Middle Teaching Building 203)

This course will talk about some global aspects of surface geometry in Euclidean 3-space (as a continuation of Elementary Differential Geometry) and also rudiments of Riemannian Geometry. Most probably, the contents of both topics will be given in a staggered manner.

(Note: some parts of the lectures are not very standard for such a course, due to students with quite different levels)

Note: If you want to do some exercises, you can find some in related sections of the following references [2], [5] and [6]

1) Rigidity of the standard 2-sphere in Euclidean 3-space;

2) Hilbert theorem: Poincare upper half plane with hyperbolic metric CANNOT be ISOMETRICALLY IMMERSED in Euclid 3-space;

3) Riemannian manifolds

(i) definition of (smooth) manifolds, (co-)tangent spaces (bundles), (smooth) vector fields, Lie bracket, affine connections;

(ii) riemannian metrics, riemannian (Levi-Civita) connections, fundamental theorem of riemannian geometry;

(iii) curvature tensor and its properties (in particular the first Bianchi identity), sectional curvature, ricci curvature and scalar curvature;

*(iv) (r,s)-type tensors, covariant differentiation (with respect to an affine connection); the second Bianchi identity of the curvature tensor;

*(v) differential form, exterior differentiation, dual of exterior differentiation, laplace-beltrami operator, harmonic forms (functions); an introduction of de Rham and Hodge theorems,.....

(vi) isometry, isometric immersion (imbedding), (riemannian) submanifolds; normal bundle, induced connections; the second fundamental form, totally geodesic submanifolds, mean curvature (vector);

4) parallel translation of vector fields (along a curve, with respect to an affine connection); geodesics, exponential map, geodesic polar coordinates (Gauss lemma), (local) minimality of geodesics, geodesic convex neighborhoods;

5) (geodesic, metric) completeness, Hopf-Rinow theorem

6) Hopf theorem (constant mean curvature (immersed) surfaces of 0-genus in Euclidean 3-space)

(i) isothermal parameters, existence; complex (conformal) structure; Riemann surfaces

(ii) holomorphic quadratic differentials on a Riemann surface

(iii) Hopf theorem: the equations of motion and structure of surfaces under isothermal parameters; Hopf‘s differential

* (iv) Wente‘s counterexample for the 1-genus case

7) the first and second variational formulae of arc length and applications: Bonnet-Myers theorem, Weinstein theorem, Synge theorem

8) Jacobi fields, conjugate points, cut points and cut locus

9) Cartan-Hadamard theorem and Space forms

10) Index form; comparison theorems (Rauch, Hessian, Laplace, Volume)

11) Applications of comparison theorems (Laplace, Volume): Bochner formulae and another proof of Lapace comparison theorem; the splitting theorem of Cheeger-Gromoll for manifolds of nonnegative Ricci curvature; the maximal diameter theorem

References

[1]J. Cheeger & D. Ebin, Comparison theorems in Riemannian Geometry, AMS Chelsea Publishing, 1975

[2] M. do Carmo,Differential Geometry of Curves and Surfaces, Dover Publications, INC., revised & updated second edition, 2016:Chapter 5.

[3] S. Donaldson,Riemann Surfaces, Oxford University Press.

[4] H. Hopf,Differential Geometry in the Large, Springer-Verlag, 1983:Part II.

[5] J. Jost,Riemannian Geometry and Geometric Analysis, Springer, Seocnd ed., 1998:Chapter 3, 4.

[6] P. Petersen,Riemannian Geometry, Springer, Second edition.

[7] 伍鸿熙,黎曼几何初步, 北京大学出版社, 1989.

[8] 忻元龙,黎曼几何讲义, 复旦大学出版社, 2010.

2.Introduction to Metric Riemannian Geometry(mini-course, by Professor Xiaochun Rong, Rutgers;May 12-23, 2014)

Abstract: The purpose of the mini course is to give a quick introduction to one of the important subjects in Metric Riemannian Geometry: geometric and topological structures on manifolds with Ricci curvature bounded below. We will introduce basic analytic and geometric tools, and using which we will prove most classical results in the subject. We will also extend the discussion to recent advances. This course will cover the following three topics:

1)Ricci Curvature Comparison and Applications

2)Gromov-Hausdorff Topology

3)Degeneration of Metrics with Ricci Curvature bounded Below (which likely exclude some details due to a time constraint)

Prerequisite: Basic knowledge on Riemannian geometry (Riemannian metrics, connections, curvature, geodesics, variation formulae, etc), and basic knowledge on Topology (set topology, covering spaces, fundamental groups, etc).

1.Riemannian Geometry(Winter of 2013 and Spring of 2014): This is a course on Riemannian Geometry for graduate students in geometry. Topics mainly include: Riemannian metrics, fundamental theorem of Riemannian geometry (Levi-Civita connection), curvature tensor (sectional curvature, Ricci curvature), geodesic (exponential map, geodesic convex neigborhood, Gauss lemma), completeness (Hopf-Rinow theorem), Jacobi fields and conjugate points, totally geodesic submanifolds, Cartan-Hadamard theorem, space forms, the first and second variational formulae for geodesic (Bonnet-Myers theorem, Synge Theorem, index form), cut locus, comparison theorems (Rauch comparison theorem, Hessian comparison theorem, Laplace comparison theorem, Cheeger-Gromoll splitting theorem, Bishop-Gromov volume comparison theorem, Toponogov comparison theorem).

9. (with Huihong Jiang)Manifolds of positive Ricci curvature, quadratically asymptotically nonnegative curvature, and infinite Betti numbers. Preprint, arXiv: 1905.01616 math.DG SCIENCE China Mathematics, 2021.

8. (with Huihong Jiang)Manifolds of positive Ricci curvature withquadratically asymptotically nonnegative curvatureand infinite topological type, Communications in Analysis and Geometry,Volume 29， No. 5, 1233-1253, 2021.

7. (with Huihong Jiang)Diameter growth and bounded topology of complete manifolds with nonnegative Ricci curvature,Ann Global Anal. Geom.51(2017), no.4,359–366.

6. (with Yi Zhang)A new proof of a theorem of Petersen,SCIENCE CHINA, Mathematics,59(2016), no. 5, 935-944.

5. (with J. Jost and K. Zuo)Harmonic maps and singularities of period mappings,Proc. Amer. Math. Soc.143(2015), 3351-3356.

4. (with J. Jost and K. Zuo)Harmonic metrics on unipotent bundles over quasi-compact Kaehler manifolds.Preprint, 2009

3. (with Qihua Ruan and Jiaxian Wu)Gradient estimates for a nonlinear diffusion equation on complete manifolds,Chinese Annals of Mathematics,Ser. B 36(2015), no. 6, 1011-1018.

2. (with Jiaxian Wu)Gradient estimates and Harnack inequality for a nonlinear parabolic equation on complete manifolds,Communications in Mathematics and Statistics,1(2013), no.4, 437-464.

1. (with Qihua Ruan and Jiaxian Wu)Gradient Estimate for Exponentially Harmonic Functions on Complete Riemannian Manifolds,Manuscripta Mathematica,143(2014), no. 3-4, 483-489.

Time: Dec. 31, 2021 (Friday) 14:30-15:30, Tencent meeting 663 460 358, code 211231

Prof. Xiaochun Rong (Rutgers University): Collapsed manifolds with local Ricci bounded covering geometry.

Abstract: We will survey some recent progress on the study of collapsed manifolds with local Ricci

bounded covering geometry.

Time: Thursday 15:00-16:30, Place: Science Buildings, No. 6, Rm 901

Dr Kewei Zhang (BICMR, Peking University):Recent progress on Tian‘s partial C^0 estimate (I, II)

2019.10.31,15:00-16:00; 2019.11.1,10:30-11:30(Middle Lecture Room, 703)

Abstract:Partial C^0 estimate plays crucial roles in the proof of Yau-Tian-Donaldson conjecture, which measures the very ampleness of a line bundle in a quantitative way. I will report some recent progress on the partial C^0 estimate. For instance, I will show that, along the normalized Kaehler-Ricci flow on a Fano manifold, the partial $C^0$ estimate holds uniformly. I will also show that for polarized Kaehler manifolds with Ricci lower bound and diameter upper bound, the Bergman kernel has a uniform polynomial growth, which improves the previous result of Donaldson-Sun and Liu-Szekelyhidi.

Prof. Akito Futaki (YMSC, Tsinghua):Coupled Kaehler-Einstein metrics (Colloquium)

2019.10.18, 15:00-16:00(Large Conference Room, 706)

abstract : Coupled Kaehler-Einstein metrics consist of Kaehler metrics on Fano manifolds satisfying a system of Monge-Amepere equations extending the Kaehler-Einstein equation. There are obstructions naturally extending those for Kaehler-Einstein metrics. We show a residue formula for one of the obstructions and a concrete example of the computation. This talk is based on joint works with Yingying Zhang.

In the colloquium talk, I will start from the classical results of Yau and Aubin, so it should be understandable for general audience.

Prof. Akito Futaki (YMSC, Tsinghua): Conformally Kaehler, Einstein-Maxwell metrics

2019.10.17, 15:00-16:00(Middle Lecture Room, 703)

Abstract: A conformally Kaehler, Einstein-Maxwell (cKEM for short) metric is a Hermitian metric with constant scalar curvature on a compact complex manifold such that it is conformal to a Kaehler metric with conformal factor being a Hamiltonian Killing potential. Fixing a Kaehler class, we characterize such Killing vector fields whose Hamiltonian function with respect to some Kaehler metric in the fixed Kaehler class gives a cKEM metric. The characterization is described in terms of critical points of certain volume functional. The conceptual idea is similar to the cases of Kaehler-Ricci solitons and Sasaki-Einstein metrics since the derivative of the volume functional gives rise to a natural obstruction to the existence of cKEM metrics. This talk is based joint works with Hajime Ono.

Prof. Qi S. Zhang (UC, Riverside): A few properties of global solutions of the heat equation on Euclidean space and some manifolds

2019.09.19, 15:00-16:00(Middle Lecture Room, 703)

Abstract: We report some recent results on Martin type representation formulas for ancient solutions of the heat equation and dimension estimates of the space of these solutions under some growth assumptions.

Huihong Jiang (SJTU):

2019.06.06, 13, 20, 15:00-16:30

Ruobing Zhang (Stony Brook): Collapsed Einstein spaces: singular behaviors, geometric structures and new constructions (I,II,III,IV)

2019.04.25, 26,15:00-16:30; 2019.04.28, 15:00-16:30, 18:30-20:00　

Abstract: This mini-course concerns the geometry of collapsed Einstein manifolds which consists of four primary parts.
In my first talk, we will introduce some basic knowledge and the motivations to study the convergence theory in Riemannian geometry. Our particular focus is to study a family of Einstein spaces converging to some lower dimensional metric space, which is called "collapsing" in the literature. In addition to study the fundamental tools in understanding Einstein manifolds, we will also introduce both historical and new examples of collapsed Einstein spaces.
I will try to make the mini-course self-contained.

Philipp Reiser (KIT/SJTU): Spaces and Moduli Spaces of Riemannian Metrics with Positive Scalar Curvature (I,II,III,Ⅳ)

2019.03.21,28; 2019.04.04,11

Abstract: In which ways can a given smooth manifold be curved in a specific manner? This question is of fundamental interest in Riemannian geometry and leads to the consideration of the space of all Riemannian metrics on this manifold which satisfy the desired curvature condition. The moduli space is then obtained by identifying Riemannian metrics which are isometric and the goal is to understand the topology of these spaces. In these talks I will give an overview on the situation if one considers Riemannian metrics with positive scalar curvature on closed manifolds whose dimension is at least 5. For that I will introduce the required tools from spin geometry and surgery theory.