CIS 610, Spring 2008

Advanced Geometric Methods in Computer Science

This version (Spring 2008) will be devoted mostly to

Manifolds, Lie Groups, Lie Algebras and
Riemannian Geometry, with Applications to
Medical Imaging and Surface Reconstruction

One of our main goals will be to build enough foundations to understand some recent work in
Diffusion tensors and Shape statistics (in medical imaging).

In particular, among our goals, we aim to discuss various papers of the Asclepios Research Group headed by Nicholas Ayache.

** Asclepios, INRIA Sophia Antipolis   (html)

In particular:
** Vincent Arsigny's dissertation:
Processing Data in Lie Groups: An Algebraic Approach. Application to Non-Linear Registration and Diffusion Tensor MRI
(html)   (pdf)
** Arsigny, Fillard, Pennec, Ayache:
Geometric Means in a Novel Vector Space Structure on Symmetric Positive-Definite Matrices.
: (pdf)
** Arsigny, Fillard, Pennec, Ayache:
Log-Euclidean Metrics for Fast and Simple Calculus on Diffusion Tensors
: (pdf)
** Arsigny, Pennec, Ayache:
Polyrigid and Polyaffine Transformations: a Novel Geometrical Tool to Deal with Non-Rigid Deformations - Application to the registration of histological slices
: (pdf)
** Fillard, Pennec, Arsigny, Ayache:
Clinical DT-MRI Estimation, Smoothing and Fiber Tracking with Log-Euclidean Metrics
: (pdf)
** Xavier Pennec:
Statistical Computing on Manifolds for Computational Anatomy
: (html)   (pdf)
** Xavier Pennec:
Intrinsic Statistics on Riemannian Manifolds: Basic Tools for Geometric Measurements.
: (pdf)
** Xavier Pennec, P. Fillard and N. Ayache:
A Riemannian Framework for Tensor Computing
: (pdf)
** Ayache, Clatz, Delingette, Malandain, Pennec, Sermesant:
Asclepios: a Research Project at INRIA for the Analysis and Simulation of Biomedical Images
: (pdf)
** Durrleman, Pennec, Trouve, Ayache:
Measuring Brain Variability via Sulcal Lines Registration: a Diffeomorphic Approach
: (pdf)

Course Information
February 25, 2008


Towne 315, M,W, noon-1:30pm


Jean H. Gallier, GRW 476, 8-4405,

Office Hours: , or TBA


Basic Knowledge of linear algebra and geometry (talk to me).


There will be no official textbook(s) but I will use material from several sources including my book (abbreviated as GMA)


Problem Sets (3 or 4), project(s), or presentation.

A Word of Advice :

Expect to be held to high standards, and conversely! In addition to transparencies, I will distribute lecture notes. Please, read the course notes regularly, and start working early on the problems sets. They will be hard! Take pride in your work. Be clear, rigorous, neat, and concise. Preferably, use a good text processor, such as LATEX, to write up your solutions. You are allowed to work in small teams of at most three. We will have special problems sessions, roughly every two weeks, during which we will solve the problems together. Be prepared to present your solutions at the blackboard. I am hard to convince, especially if your use blatantly ``handwaving'' arguments.

Brief description:

This course covers some basic material on manifolds, Lie groups Lie algebras and Riemannian Geometry, keeping in mind applications of these theories to medical imaging, computer vision and machine learning. The treatment will be rigorous but I will try very hard to convey intuitions and to give many examples illustrating all these concepts.

Tentative Syllabus

Next semester (Spring 2008), I intend to cover (1)-(8) below.
  1. Introduction to Lie groups, Lie algebras and manifolds
  2. Review of Groups and Group Actions
    • Groups
    • Group Actions and Homogeneous Spaces, I
    • Topological Groups
  3. Manifolds, Tangent Spaces, Cotangent Space
    • Manifolds
    • Tangent Vectors, Tangent Spaces, Cotangent Spaces
    • Tangent and Cotangent Bundles, Vector Fields
    • Submanifolds, Immersions, Embeddings
    • Integral Curves, Flow, One-Parameter Groups
    • Partitions of unity
    • Orientation of manifolds
  4. 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
  5. Riemannian Manifolds and Connections
    • Riemannian metrics
    • Connections on manifolds
    • Parallel transport
    • Connections compatible with a metric; Levi-Civita connections
  6. 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
  7. 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
  8. Metrics and curvature on Lie groups
    • Left and right invariant metrics
    • Bi-invariant metrics
    • Connections and curvature of left-invariant metrics
    • The Killing form

Additional References:

Lie Groups:

Lie groups, Lie algebras, and representations, Hall, Brian, Springer (GTM No. 222)

Lie Groups. An introduction through linear Linear groups, Wulf Rossmann, Oxford Science Publications, 2002

An Introduction to Lie Groups and the Geometry of Homogeneous Spaces, Arvanitoyeogos, Andreas, AMS, SML, Vol. 22, 2003

Lectures on Lie Groups and Lie Algebras, Carter, Roger and Segal, Graeme and Macdonald, Ian, Cambridge University Press, 1995

Lie Groups, Duistermaat, J.J. and Kolk, J.A.C., Springer Verlag, Universitext, 2000

Lie Groups Beyond an Introduction, Knapp, Anthony W., Birkhauser, Progress in Mathematics, Vol. 140, Second Edition, 2002

Theory of Lie Groups I, Chevalley, Claude, Princeton University Press, first edition, Eighth printing, Princeton Mathematical Series, No. 8, 1946

Foundations of Differentiable Manifolds and Lie Groups, Warner, Frank, Springer Verlag, GTM No. 94, 1983

Introduction to Lie Groups and Lie Algebras, Sagle, Arthur A. and Walde, Ralph E., Academic Press, 1973

Representation of Compact Lie Groups, Brocker, T. and tom Dieck, T., Springer Verlag, GTM, Vol. 98, 1985

Elements of Mathematics. Lie Groups and Lie Algebras, Chapters 1--3, Bourbaki, Nicolas, Springer, 1989

Introduction a la Theorie des Groupes de Lie Classiques, Mneimne', R. and Testard, F., Hermann, 1997

Manifolds and Differential Geometry:

Differential Geometry. Curves, Surfaces, Manifolds, Wolfgang Kuhnel, AMS, SML, Vol. 16, 2002.

Riemannian Geometry, Do Carmo, Manfredo, Birkhauser, 1992.

Differential Geometry of Curves and Surfaces, Do Carmo, Manfredo P., Prentice Hall, 1976.

Geometry of Differential Forms, Shigeyuki Morita, AMS, Translations of Mathematical Monographs, Vol. 201, First Edition.

Riemannian Geometry, Sylvestre Gallot, Dominique Hulin, Jacques Lafontaine, Springer Verlag, Universitext, 2004, Third Edition.

Semi-Riemannian Geometry With Applications to Relativity, Barrett O'Neill, Academic Press, 1983, First Edition.

A Panoramic View of Riemannian Geometry, Marcel Berger, Springer, 2003, First Edition.

Geometry VI. Riemannian Geomety, M.M. Postnikov, Springer, Encyclopaedia of Mathematical Sciences, Vol. 91, 2001, First Edition.

Riemannian Geometry, Takashi Sakai, AMS, Translations of Mathematical Monographs, Vol. 149, 1997, First Edition.

Morse Theory, John Milnor, Princeton University Press, Annals of Mathematics Studies, No. 51, 1969, Third Printing.

Riemannian Geometry. A beginner's guide, Frank Morgan, A.K. Peters, 1998, Second Edition.

Geometry And Music

In mathematics, and especially in geometry, beautiful proofs have a certain ``music.'' I will play short (less than 2mn) pieces of classical music, or Jazz, whenever deemed appropriate by you and me!

Some Slides and Notes

Papers and Talks Suitable for a Project

Back to Gallier Homepage

published by:

Jean Gallier