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