## Isometric Embeddings

Department:
MATH
Course Number:
8803-GHO
Hours - Lecture:
3
Hours - Lab:
0
Hours - Recitation:
0
Hours - Total Credit:
3
Typical Scheduling:
Not typically scheduled
Prerequisites:
Interest in geometry or its potential applications, together with mathematical maturity at the level of a beginning math grad student. There will be no specific course requirements; however, some familiarity with differential geometry (e.g., Math 6455) or manifold theory (e.g., Math 6452) could be helpful. Advanced undergraduate students with some knowledge of classical differential geometry of curves and surfaces (e.g., Math 4441) may also take this class.
Course Text:
We will study various handouts, papers, and excerpts from different texts including:

— Spivak, A comprehensive Intro. to Diff, Geometry Vol. 5
— Han and Hong, Isometric Embeddings
— Eliashberg & Mishachev, Intro to h-Principle
— Pak, Lectures on Discrete and Polyhedral Geometry
— Convex Polyhedra, Alexandrov
— Extrinsic Geometry of Convex Surfaces, Pogorelov
— Papers and lecture notes of the instructor
Topic Outline:
The main aim of this class is to study the isometric rigidity of closed smooth surfaces in Euclidean space, which has been a major open problem in differential geometry dating back to Euler and Maxwell. We will go over a host of related results, relevant techniques, and auxiliary problems ranging from discrete and polyhedral geometry to the theory of partial differential equations. The results we will cover include:

— Cauchy’s theorem for rigidity of convex polyhedra
— Cohn-Vossen’s generalization of Cauchy’s theorem to smooth surfaces
— Alexandrov’s existence theorem for isometric embeddings of convex surfaces
— Pogorelov’s uniqueness theorem for convex surfaces
— Stoker’s results on rigidity of non convex polyhedra
— Isometric embeddings via Gromov’s h-Principle, including Nash’s C^1 embedding theorem
— Nirenberg’s problem on rigidity of negatively curved tubes
— Yau’s problem on positively curved surfaces with boundary