CIS 610, Spring 2023
Brief description:
This course covers
some basic material on manifolds,
Riemannian metrics, Lie groups, Lie algebras,
and homogeneous manifolds,
keeping in mind applications of
these theories to machine learning,
computer vision and robotics.
The treatment will be rigorous but
we will try very hard to convey intuitions and to give many
examples illustrating all these concepts.
Syllabus:

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

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, OneParameter Groups

Abstract manifolds (charts, atlases, etc);

Partitions of unity

(*) Orientation of manifolds

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 GroupsLie Algebras

Riemannian Manifolds and Connections

Riemannian metrics

Connections on manifolds

Parallel transport

Connections compatible with a metric; LeviCivita connections

Geodesics on Riemannian Manifolds

Geodesics, local existence and uniqueness

The exponential map

Complete Riemannian manifolds, HopfRinow Theorem, CutLocus

The calculus of variation applied to geodesics

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

Metrics and curvature on Lie groups

Left and right invariant metrics

Biinvariant metrics

Connections and curvature of leftinvariant metrics

The Killing form

Semisimple Lie algebras; a glimpse at the classification theorem

Riemannian submersions

Homogeneous manifolds

Reductive homogeneous manifolds

Symmetric spaces

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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