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

Instructor: |
Facundo Mémoli, m e m o l i @ m a t h . o s u . e d u |

Course code: | Math 6702 -- Spring 2015 |

Times: |
MWF -- 1.50--2.45 |

Location: |
Dulles Hall 020. Map |

Topics: | Riemannian manifolds, distance, volume; covariant derivatives, Levi-Civita connection; geodesics, curvatures: sectional, Ricci, scalar; Jacobi fields, conjugate points; Rauch comparison, Hopf-Rinow, Hadamard, Preissman's and Berger's Sphere 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 background reading. |

Books: The book by Gallot, Hulin & LaFontaine ([GHL] from now on) is a good resource. I may 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]). | |

Assumed background. To read on your own: Chapter 1 of [GHL] and Chapters 1 to 4 of Do Carmo's Differential Geometry of Curves and Surfaces. | |

Lecture 1 (Jan. 12th). $(M,g^M)\in\mathcal{R}$ 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 (Jan 14th). Length of curves. RMs as metric spaces. | |

Lecture 3 (Jan. 16th). Energy of curves, relationship to Length. Isometries of metric spaces vs. Riemannian isometries -- the Myers-Steenrod theorem. | |

Lecture 4 (Jan. 21st). Connections. The Levi-Civita connection. | |

Lecture 5 (Jan. 23rd). Expression in local coordinates. Christoffel symbols. Covariant derivative along a curve. Parallel translation. Gradient. | |

Lecture 6 (Jan. 26th) Covariant derivative of tensors. Hessian of a function. Laplace-Beltrami operator. The flat case. | |

Lecture 7 (Jan. 28th). Geodesics. Geodesic equation. | |

Lecture 8 (Jan. 30th). The exponential map. Normal coordinates. Gauss Lemma. | |

Lecture 9 (Feb. 2nd). Student presentation: Sunhyuk Lim -- Systolic Geometry. | |

Lecture 10 (Feb. 4th). Student presentation: Corey Staten -- The Myers-Steenrod theorem. | |

Lecture 11 (Feb. 6th). Student presentation: Irfan Glogic -- Classification of Flat Tori. | |

Lecture 12 (Feb. 9th) First variation of length. Geodesic completeness. Hopf-Rinow theorem. | |

Lecture 13 (Feb. 11th) The Riemann curvature Tensor. | |

Lecture 14 (Feb. 13th) Sectional, Ricci, and Scalar curvature. | |

Lecture 15 (Feb. 16th) Spaces of constant curvature. | |

Lecture 16 (Feb. 18th) First and second Variation of Energy and Length. | |

Lecture 17 (Feb. 20th) Jacobi fields (I). | |

Lecture 18 (Feb. 23rd) Jacobi fields (II). Geodesic variations. | |

Lecture 19 (Feb. 25th) Geodesic variations. Taylor Expansion of the metric. | |

Lecture 20 (Feb. 27th). Interpretation of Jacobi fields. Conjugate points. | |

Lecture 21 (Mar. 2nd). Student presentation: Jimmy Voss -- Geodesics on $P^n(\mathbb{C})$. | |

Lecture 22 (Mar. 4th). Student presentation: Waylon Chen -- Metrics on Lie Groups. | |

Lecture 23 (Mar. 6th). Student presentation: Samir Chowdhury -- Model spaces: their geodesics and sectional curvatures. | |

Lecture 24 (Mar. 9th). Gauss Lemma and applications. | |

Lecture 25 (Mar. 11th). The index form. Jacobi theorems. | |

Lecture 26 (Mar. 13th). Bonnet-Myers theorem. | |

Lecture 27 (Mar. 23rd). Hadamard theorem. Morse-Schoenberg theorem. Rauch I Comparison Theorem for Jacobi fields. | |

Lecture 28 (Mar. 25th). Statement of Volume comparison theorems. Cheng's maximal diameter theorem. | |

Lecture 29 (Mar. 27th). Rauch II. | |

Lecture 30 (Mar. 30th). Metric tensor on manifolds of constant curvature. | |

Lecture 31 (Apr. 1st). Volume of metric balls in manifolds of constant curvature. Multidimensional variations. | |

Lecture 32 (Apr. 3rd). Matricial Jacobi equation. Volume of metric balls. | |

Lecture 33 (Apr. 6th). Bishop-Gunther theorem. | |

Lecture 34 (Apr. 8th). Bishop volume comparison theorem. | |

Lecture 35 (Apr. 10th). Gromov's relative volume comparison theorem. Gromov's precompactness theorem. | |

Lecture 36 (Apr. 13th). Gromov's precompactness theorem and the GH distance. | |

Lecture 37 (Apr. 15th).. | |

Lecture 38 (Apr. 17th).. | |

Lecture 39 (Apr. 20th).. | |

Lecture 40 (Apr. 22th).. | |

Lecture 41 (Apr. 24th).. | |

Lecture 42 (Apr. 27th).. |