CIS 610, Spring 2018

Brief description:

Syllabus:

1. The matrix exponential
   • The surjectivity of exp: so(n) -> SO(n)
   • Rodrigues formula
   • The surjectivity of exp: su(n) -> SU(n)
   • Symmetric positive definite matrices (SPD).
   • The bijection exp: S(n) -> SPD
2. Manifolds, Tangent Spaces, Cotangent Space
   • Introduction to manifolds
   • Manifolds in R^n (charts, etc.)
   • Tangent Vectors, Tangent Space, Cotangent Spaces
   • Vector Fields
   • Submanifolds, Immersions, Embeddings
   • Integral Curves, Flow, One-Parameter Groups
   • Abstract manifolds (charts, atlases, etc);
   • Partitions of unity
   • (*) Orientation of manifolds
3. Lie Groups, Lie Algebra, Exponential Map
   • Lie Groups and Lie Algebras
   • Left and Right Invariant Vector Fields, Exponential Map
   • Homomorphisms, Lie Subgroups
   • The Correspondence Lie Groups--Lie Algebras
4. Riemannian Manifolds and Connections
   • Riemannian metrics
   • Connections on manifolds
   • Parallel transport
   • Connections compatible with a metric; Levi-Civita connections
5. Geodesics on Riemannian Manifolds
   • Geodesics, local existence and uniqueness
   • The exponential map
   • Complete Riemannian manifolds, Hopf-Rinow Theorem, Cut-Locus
   • The calculus of variation applied to geodesics
6. Curvature in Riemanian Manifolds
   • The curvature tensor
   • Sectional curvature
   • Ricci curvature
   • Isometries and local isometries
   • Riemannian covering maps
   • (*) The second variation formula and the index form
   • (*) Jacobi fields
   • (*) Applications of Jacobi fields and conjugate points
   • Cut locus and injectivity radius: some properties
7. Metrics and curvature on Lie groups
   • Left and right invariant metrics
   • Bi-invariant metrics
   • Connections and curvature of left-invariant metrics
   • The Killing form
   • Semisimple Lie algebras; a glimpse at the classification theorem
8. Riemannian submersions
9. Homogeneous manifolds
   • Reductive homogeneous manifolds
   • Symmetric spaces
10. Specific Groups, Manifolds and Applications
    • SO(3), SE(3), RP^3 (Vision)
    • Stiefel manifolds S(m,n)
    • Grassmann manifolds G(m,n) (learning)

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Jean Gallier