Cartan Geometry Seminar

Fall-Winter 2006-07

Department of Mathematics and Statistics, Dalhousie University

Seminar organizer Roman Smirnov

Location (unless otherwise indicated): Room 107 (Chase Building); Time (unless otherwise indicated): Friday 1:00pm-2:30pm
The main objective of the seminar is to study the approach to differential geometry developed by Elie Cartan. The approach is based on the theory of Lie groups and moving frames.
Previous years seminars on line: 2005-06

To retrieve the info on a specific seminar, please click on the corresponding date:

15.09.2006, 20.09.2006, 22.09.2006, 29.09.2006, 6.12.2006,

References:

  • E. Cartan, Riemannian Geometry in an Orthogonal Frame (translated by V. V. Goldberg), World Scientific, 2001;

  • S. S. Chern, W. H. Chen, K. S.Lam, Lectures on Differential Geometry, World Scientific, 1998;

  • T. I. Ivey and J. M. Landsberg, Cartan for Beginners: Differential Geometry via Moving Frames and Exterior Differential Systems, Graduate Studies in Mathematics 61, AMS, 2003;

  • P. J. Olver, Classical Invariant Theory, London Mathematical Society Student Texts 44, Cambridge University Press, 1999;

  • R. W. Sharpe, Differential Geometry: Cartan's Generalization of Klein's Erlangen Program Graduate Texts in Mathematics, Springer, 1996.

     

    Date: September 15, 2006 (Friday)
    Speaker: Roman Smirnov
    Title: Lie groups, moving frames and the geometry of Euclidean space I.
    Abstract: I will briefly review the main underlying ideas of the Cartan geometry as applied to the geometry of the Euclidean space, including moving frames in Euclidean space, Maurer-Cartan structure equations, Serret-Frenet equations in moving frames.

     

    Date: September 20, 2006 (Wednesday) - Honours Seminar
    Speaker: Roman Smirnov
    Title: The geometry via Lie groups and moving frames: An introduction to Elie Cartan's philosophy
    Abstract: ``When we do analytical Euclidean geometry, we would prefer an orthogonal coordinate system, instead of a general Cartesian system. Cartan carries this out for Riemannian geometry. In this sense the book does need any further recommendation.'' (S.-S. Chern (1911-2004), from his Foreword to ``Riemannian geometry in an orthogonal frame'' by Elie Cartan.)

     

    Date: September 22, 2006 (Friday)
    Speaker: Roman Smirnov
    Title: Lie groups, moving frames and the geometry of Euclidean space II.
    Abstract: We will discuss how hypersurfaces in Euclidean space can be naturally studied in terms of the associated Darboux frames.

     

    Date: September 29, 2006 (Friday)
    Speaker: Roman Smirnov
    Title: Lie groups, moving frames and the geometry of Euclidean space III.
    Abstract: I will discuss Cartan's lemma, the first and second fundamental forms, Gauss and Godazzi equations in moving frames.

     

    Date: December 6, 2006 (Wednesday, 12:30pm-1:30pm, Colloquium Room-Chase 319)
    Speaker: Joshua Horwood (University of Cambridge)
    Title: Separation of variables and orthogonal coordinate webs in pseudo-Riemannian manifolds
    Abstract: In this talk, I shall present a purely algebraic method of determining orthogonally separable coordinate systems and first integrals for natural Hamiltonians on three-dimensional Minkowski space. In addition, I'll show that analogous results for the Minkowski plane, hyperbolic space and De Sitter space, the homogeneous subspaces of Minkowski space, are essentially obtained for "free". The method is based on the invariants of Killing tensors which characterize the associated orthogonal webs under the action of the isometry group. I will outline an invariant classification scheme for the corresponding 39 orthogonal coordinate webs in Minkowski space, emphasizing not only the role of the group invariants in its development, but also the importance of group covariants, reduced invariants and conformal symmetries. Applications of the method will follow.