Event

This course is an introduction to modern differential geometry. Differential geometry is the study of geometry of spaces using analytic and linear algebraic methods. After laying down the foundational definitions and basic properties of smooth manifolds, tangent vectors, and Riemannian metrics, we will develop notions of linear connections and covariant derivatives allowing us to do differential calculus on these manifolds. We will continue our journey of understanding the shape of these manifolds by developing concepts of curvature tensors, geodesics, parallel transport and Jacobi fields. We will also cover the celebrated Bonnet-Myers and Cartan-Hadamard theorems which show us that curvature conditions on a manifold can to some extent dictate the geometry and topology of the manifold.