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

9) Cartan-Hadamard theorem and Space forms

10) comparison theorems (Hessian, Rauch, Laplace, Volume)

**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 edition, 1998: **Chapter 3, 4**.

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

[7] Y.-L. Xin, *Lectures on Riemannian Geometry* (in Chinese), Fudan University Press.

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).