S. S. Chern Lecture Series, Beijing, China September 22, 2011

Plan of Lecture The lecture will have four parts: Part 1: History and a general description of the subject. Part 2: The Weyl embedding problem. Part 3: Geometric variational problems; Plateau and Yamabe problems. Part 4: Geometric flows; Ricci and mean curvature flow. This lecture is not intended to be a list of the most important developments in the subject since 1950. The selection is based on my interests. I have not mentioned complex geometry, so have omitted Yau’s solution of the Calabi Conjecture and the subsequent development in String Theory. I have also omitted the Atiyah-Singer Index Theorem. Both of these belong on any list of the most important developments.

Part 1: Brief History of Differential Geometry Differential Geometry begins with curves in the plane. Greek geometry, as exposited in Euclid’s Elements, focused on lines and circles in the plane and space.

Curvature of Curves

Curves whose curvature varies from point to point entered into geometry and physics beginning in the 14th century with important contributions by N. Oresme, J. Kepler, I. Newton, and L. Euler.

The curvature at a point is the reciprocal of the radius of the circle that best fits the curve at the point.

Curves Euler developed much of the modern theory of plane curves.

Curved Surfaces I A surface is an infinitely thin two dimensional membrane in space.

M¨obius Band

Torus

Higher Genus Surfaces

Curved Surfaces II The key defining property of a surface is that we can introduce (two) local coordinates near any point of a surface.

Geometry of Surfaces I The Differential Geometry of surfaces was developed by C. F. Gauss around 1825.

Geometry of Surfaces II Gauss defined the curvature of a surface in terms of curvatures of curves on the surface.

Intrinsic Geometry Certain aspects of the geometry of a surface are determined by measurements made along the surface without reference to the larger three dimensional space. Such quantities are called intrinsic, and these include lengths of curves on the surface, angles between curves, and areas of regions on the surface. Gauss’ most remarkable discovery, which he called his Theorema Egregium, is the result that a certain expression in the curvatures of a surface is intrinsic. This expression is K = k1 k2 , and it is called the Gauss curvature of the surface. The proof involved a very difficult calculation. This motivated the study of intrinsic geometries or metrics on a surface which do not necessarily come from embeddings into three space. These are measurements of distances and angles for which the Pythagorean theorem holds approximately for small right triangles.

Higher Dimensions The intrinsic geometry of Gauss was generalized to higher dimensions by B. Riemann in 1854.

Sectional Curvature Just as the curvatures of a surface can be understood in terms of curvatures of certain curves on the surface, the curvatures of higher dimensional spaces (called Riemannian manifolds) can be understood in terms of the Gauss curvatures of certain surfaces passing through a point. These Gauss curvatures are called sectional curvatures and there is one for each two dimensional plane at a point.

Part 2: The Weyl Embedding Problem I

A two dimensional geometry may not be embedded in a natural way in space. For example, the hyperbolic plane is a geometry of Gauss curvature equal to −1, and while pieces of it can be embedded in space, a theorem of Hilbert asserts that it cannot be globally realized in space. In 1916 H. Weyl conjectured and outlined a proof that a closed surface of positive Gauss curvature can be isometrically embedded in space as the boundary of a convex body. He was not able to complete the