Differential Geometry

From "A Journey into Gravity and Spacetime" by Wheeler.

Course Info

Instructor: Facundo Mémoli,   m e m o l i @ m a t h . o s u . e d u
Course code: Math 6702 -- Spring 2018
Times: MWF 1.50--2.45
Location: TBA.
Topics: Riemannian manifolds, distance, volume; covariant derivatives, Levi-Civita connection; geodesics, curvatures: sectional, Ricci, scalar; Jacobi fields, first and second variation of theenergy functional, conjugate points, Rauch comparison, Hopf-Rinow, Hadamard, Bishop and Bishop-Gromov theorems. Other topics: Spectral geometry, families of manifolds with bounded geometry.
Assessment: TBA.
Prerequisites: Some degree of familiarity with differentiable manifolds and differential geometry of curves and surfaces. See below for required background knowledge.

Lectures and resources

Assumed background. To read and digest on your own before the first day of class: Chapter 1 of [GHL] and Chapters 1 to 4 of Do Carmo's Differential Geometry of Curves and Surfaces.
Books: The book by Gallot, Hulin & LaFontaine ([GHL] from now on) is a good resource. I will also use parts of Sakai's book ([S] from now on), and Petersen's book ([P] from now on). The book by Berger "A panoramic view of Riemannian Geometry" ([B]) is a very good read. The following paper by Petersen is a very good summary of some global results in RG: "Aspects of global Riemannian Geometry" ([AGRG]).
Lecture 0 (M Jan. 8th). Welcome, contents, syllabus, organization.
Lecture 1 (W Jan. 10th). $(M,g^M)\in\mathcal{R}$ and maps $F:M\rightarrow N$ between RMs. Metric tensors in \mathbb{R}^n. Riemannian metrics on manifolds. Existence of a Riemannian metric. Some examples.
Lecture 2 (F Jan 12th). Length of curves. RMs as metric spaces.