Introductory Riemannian Geometry

Overview
I intend on giving a miniseries of seminars introducing some basic results on global Riemannian geometry.

Topics

 * Historical blurb
 * Differential manifold
 * Definitions, submersion and immersion
 * Exterior calculus
 * de Rham cohomology groups and de Rham theorem
 * Vector bundles
 * Vector bundles and principal G-bundles
 * connexion on vector bundles, induced and pull-back
 * curvature of a connexion, Bianchi identities
 * Riemannian manifold
 * definition, Levi-Civita connexion
 * the exponential map and geodesics
 * the Laplacian and Hodge theorem
 * Riemannian immersion and the second fundamental form
 * Jacobi fields, conjugate points
 * Hopf-Rinow, cut locus
 * Curvature in Riemannian geometry
 * sectional, Ricci, scalar curvature
 * Hadamand-Cartan
 * Bonnet-Myers
 * Synge-Weinstein
 * 1/4-pinching
 * Volume comparison theorems
 * Ricci curvature and growth of groups
 * Bochner formula and Bochner theorem
 * Chern-Gauss-Bonnet
 * Toponogov theorem and Betti numbers
 * Lines and Cheeger-Gromoll splitting theorem
 * Other topics if time permits
 * 8 model geometries of 3-manifolds
 * bumpy metrics
 * Homogeneous and Symmetric spaces, holonomy classification
 * Spectral geometry
 * Riemannian orbifolds
 * Mini-Twister theory in 3-manifold
 * ASD connexions on 4-manifold
 * calibrations