- 杰出教师 OUTSTANDING FACULTY
- 博士后 POSTDOCS
- 2020年博士生导师 SUPERVISOR FOR 2020 PH.D. PROGRAM
- 2020年统计学交叉平台博士生导师 SUPERVISOR FOR 2020 STATISTICS CROSS-PLATFORM PROGRAM

师资队伍

FACULTY

杨义虎
Yihu Yang

教授
Professor

办公室 Office：

6 号楼 905

办公接待时间 Office Hour：

办公室电话 Office Phone：

54740171

E-mail：

yangyihu at sjtu.edu.cn

教育背景 Education：

博士，1994，复旦大学

Ph.D., 1994, Fudan University

研究兴趣 Research Interests：

黎曼几何，复几何，几何分析

Riemannian Geometry, Complex Geometry, Geometric Analysis

教育背景/经历 Education

Ph.D. Institute of Mathematics, Fudan University, 1994

M.S. Institute of Mathematics, Fudan University, 1991

B.S. Department of Mathematics, Lanzhou University, 1988

工作经历 Work Experience

2012- Shanghai Jiao Tong University

1996-2013 Tongji University

1994-1996 Institute of Applied Mathematics, CAS

Courses

**11. Geometry and Topology of Manifolds **(周四, 3-5节, 陈瑞球楼216)

1. Overview of vector bundles and Riemannian manifolds

1.1. Vector bundles and connections

1.2. Rudiments of Riemannian manifolds

2.

**...............**

**References**

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

[2] P. Li, Geometric Analysis,

[3] H-H Wu, The Bochner technique in Differential Geoemtry

**.............**

**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. Hilbert theorem

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:** 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

..........

[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] 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

[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 manifolds** **and 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 are **NOT** in the teaching plan.

**Chapter 1** **Curves 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 readings** for 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 2** **Surfaces 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. 3 ****Intrinsic 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 manifolds** **and 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 are **NOT** in the teaching plan.

**Chapter 1** **Curves 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 readings** for 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 2** **Regular (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. 3 ****Intrinsic 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 4** **Selected topics on the global aspects of intrinsic 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)

***Chapter 5 Regular (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).

Some recent preprints, publications

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

8. (with Huihong Jiang) **Examples of manifolds of positive Ricci curvature with quadratically nonnegatively curved infinity and infinite topological type**. Preprint, Nov. 08, 2017;

**Manifolds of positive Ricci curvature withquadratically asymptotically nonnegative curvatureand infinite topological type** (revised version, Nov. 06, 2018), Communications in Analysis and Geometry*, *to appear.

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.

Geometric Seminar

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)**

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.

We will also present a new observation on the time analyticity of solutions of the heat equation under natural growth conditions. One application is a if and only if solvability condition of the backward heat equation, i.e. under what condition can one turn back the clock in a diffusion process.

Part of the results are joint work with Fanghua Lin and Hongjie Dong.

**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.

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