Overview Edit

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

Topics Edit

  • 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

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