Description: This course will be a basic graduate introduction to differential geometry. Tentatively, we will start with some foundational material: defining manifolds, tangent bundle, vector bundles, the inverse and implicit function theorems, immersions and submersions, submanifolds, ideas of transversality. Then we will cover differential forms, integration on manifolds, Stokes' Theorem, and time permitting de Rham cohomology. After that we will get to Riemannian geometry. We will introduce and study Riemannian metrics, connections, geodesics, curvature, Jacobi fields, and the geometry of submanifolds. At the end we will try to cover some of the main classical theorems in global Riemmannian geometry, which relate curvature properties to topology, such as the Cartan-Hadamard Theorem and Myers' Theorem.
Textbook: do Carmo "Riemannian Geometry" complemented by "Differential Forms and Applications". Alternative references are John M Lee's "Introduction to Smooth Manifolds" and "Introduction to Riemannian Manifolds".
Previous semesters: Fall 2021 MATH 661, Fall 2020 MATH 742, Spring 2020 MATH 437, Fall 2019 MATH 135, Spring 2019 MATH 740, Fall 2018 MATH 868C, Spring 2017 MATH 430, Fall 2016 MATH 868D, Fall 2015 MATH 220, MATH 401, Spring 2015 MATH 461, Fall 2014 MATH 430.