


Applications of Invariant Theory to Differential Geometry / Applications de la théorie des invariants à la géométrie différentielle (Org: Robert Milson, Dalhousie University and/et Mark Fels, Utah State University)
 STEPHEN ANCO, Brock University, St. Catharines, ON
Moving frames and integrable PDE maps

In this talk I will discuss Hasimoto transformations of wave map
equations and mKdV map equations using moving frames connected with
integrable flows of curves in constant curvature Riemannian manifolds
and semisimple Lie group manifolds. Such type of transformations have
played a central role in recent analytical work on Schrodinger map
equations.
 I. ANDERSON, Utah State University, Logan, Utah
Homogeneous Spaces and Maple

In this talk I will describe current efforts to classify and analyze
low dimension homogeneous spaces using the computer algebra system
Maple. Applications to general relativity and to the exact
integration of both ordinary and partial differential equations will
be given.
 CLAUDIA CHANU, Università di Torino, via Carlo Alberto 10, 10123 Torino,
Italy
The Riemannian background of the separation of variables

The basic Riemannian structures underlying the separation of variables
in the HamiltonJacobi equation for natuaral Hamiltonian systems are
illustrated. In particular the geometrical properties of Killing
tensors and conformal Killing tensors related to the separation are
examined.
 MARK FELS, Utah State
Nonreductive pseudoRiemannian manifolds of dimension four

A method due to E. Cartan is used to give a classification of the
simply connected four dimensional nonreductive homogeneous
pseudoRiemannian manifolds.
 P. GILKEY, Math. Dept. Univ. Oregon, Eugene, OR 97403, USA
The spectral geometry of the Riemann curvature tensor

We present some examples of pseudoRiemannian manifolds which are
curvature homogeneous but not locally homogeneous. All standard local
invariants vanish for these manifolds. These manifolds are Osserman,
IvanovPetrov, and Stanilov. Some are modeled on irreducible symmetric
spaces. We also exhibit k curvature homogeneous manifolds for
arbitrarily large values of k.
 SIGBJORN HERVIK, Dalhousie University
Negatively curved leftinvariant metrics on Lie groups

In this talk we will discuss negatively curved homogeneous spaces
admitting a simply transitive group of isometries (or equivalently,
leftinvariant metrics on Lie groups). Negatively curved spaces have a
remarkably rich and diverse structure and are interesting from both a
mathematical and a physical perspective. As well as giving general
criteria for having leftinvariant metrics with negative Ricci
curvature scalar, we also consider special cases, like Einstein spaces
and Ricci nilsolitons. We point out the relevance these spaces play in
some higherdimensional theories of gravity.
 EVELYNE HUBERT, INRIA, BP 93, 06902 Sophia Antipolis, France
Constructive differential algebra for differential invariants

We study the algebraic aspects of the differential invariants as
constructed by the moving frame (Fels & Olver (1999)). Contrary to
the basic assumption of classical differential algebra (Riquier
(1910), Ritt (1951), Kolchin (1973)) the derivations naturally acting
on differential invariants satisfy some non trivial commutation
rules. We generalize classical differential algebra and differential
elimination to this setting.
This is part of a project initiated by E. Mansfield and in
collaboration with I. Kogan. The initial goal was to treat systems
that are symmetric under a Lie group action in a more appropriate
frame. Other applications are within the scope of the Maple software
developed. For instance finding a minimal set of generating
differential invariants for a given Lie group action.
 IRINA KOGAN, North Carolina State University
Noether correspondence for groupinvariant variational problems

It was shown by S. Lie that almost every variational problem,
invariant with respect to a group of point transformations, and the
corresponding EulerLagrange equations, can be written in terms of
differential invariants and invariant differential forms, and thus
reduced by the symmetry group. In this talk we define infinitesimal
variational symmetries of the reduced variational problem and show
that they lead to conservation laws for both, the reduced and the
original system of EulerLagrange equations. Computational aspects of
this approach will be also discussed.
 KAYLL LAKE, Queens University
An invariant generalization of R and T regions of spacetime

The gradient of the areal radius of an arbitrary spherically symmetric
field can be spacelike, timelike and even null. A careful distinction
of these possibilities is important and, for example, forms an
essential element of any complete proof of the Birkhoff theorem. The
possibility of spacelike and timelike gradients were studied
extensively in the Russian literature (and labeled "R" and "T"
regions respectively) some forty years ago and yet there appears to be
no readily available extension of these ideas to an arbitrary
spacetime. It is the purpose of this work to provide such an extension
and to explore this extension away from spherical symmetry. This
extension involves differential invariants of order three.
 GLORIA MARIBEFFA, University of WisconsinMadison
Invariant evolutions of curves in flat homogenous spaces

In this talk we will describe how to write geometric evolutions
(general invariant evolutions of curves) as evolutions on the Lie
algebra associated to general flat semisimple homogeneous spaces. We
will also talk about differential invariants of curves on these
spaces, moving coframes and about how to find algebraically the
evolution induced on the differential invariants by these general
geometric evolutions.
 BENJAMIN MCKAY, University of South Florida, Saint Petersburg
The Blaschke conjecture

The Blaschke conjecture aims to classify the Riemannian manifolds
whose cut loci are metric spheresin other words, spaces in which
light rays shooting out of a given point must all focus at the same
time. There is very little known about the local geometry of these
manifolds (e.g. sign of curvature is unknown), but globally we know a
few things (e.g. all geodesics are periodic). Using Cartan's method of
the moving frame, we get a simple proof that a Blaschke manifold with
the same Betti numbers as a complex projective space must be
diffeomorphic to a complex projective space.
 NICOS PELAVAS, Dalhousie University
VSI_{i} Spacetimes and the eproperty

In this talk I shall discuss Lorentzian spacetimes where all zeroth
and first order curvature invariants vanish and show how this class
differs from the VSI spacetimes. We show that for VSI spacetimes all
components of the Riemann tensor and its derivatives up to some fixed
order can be made arbitrarily small. This is illustrated by way of
some examples.
 JUHA POHJANPELTO, Department of Mathematics, Oregon State University,
Corvallis, OR 97331, USA
Differential Invariants for Pseudogroup Actions

I will report on my ongoing joint work with Peter Olver on developing
systematic and constructive algorithms for the identification and
analysis of various invariants for infinite dimensional
pseudogroups. In this talk I will focus on techniques from commutative
algebra combined with moving frames to discuss TresseKumpera type
existence results for differential invariants of a pseudogroup
action.
 VOJTECH PRAVDA, Mathematical Institute, Zitna 25, 115 67 Prague 1, Czech Republic
VSI spacetimes in higher dimensions

We determine all VSI spacetimes (i.e. Lorentzian manifolds with
vanishing curvature invariants) in arbitrary dimension higher then
four. For this reason we need to generalize some methods developed in
the context of General Relativity (e.g. the Petrov classification,
the NewmanPenrose formalism) to higher dimensions. It turns out that
the resulting VSI conditions are similar to the fourdimensional
case.
This work was done in cooperation with A. Coley and R. Milson
(Dalhousie) and A. Pravdova (Prague).
 ALENA PRAVDOVA, Mathematical Institute, Zitna 25, 115 67 Prague 1, Czech
Republic
All spacetimes with vanishing curvature invariants

A curvature invariant of order n is a scalar obtained by contraction
from a polynomial in the Riemann tensor and its covariant derivatives
up to the order n. We determine all four dimensional Lorentzian
manifolds for which all curvature invariants of all orders vanish (VSI
spacetimes). There are 16 nonflat classes of such spacetimes, one of
them being the well known ppwave. All corresponding metrics may be
given explicitly.
This work was done in cooperation with A. Coley and R. Milson
(Dalhousie) and V. Pravda (Prague).
 ROMAN SMIRNOV, Dalhousie University
A new invariant theory: Invariants, covariants and joint
invariants of Killing tensors

Multiple factors have contributed to the recent resurgence of interest
in the classical invariant theory (CIT) of homogeneous polynomials
among pure and applied mathematicians alike.
The factor that brought my collaborators and me to the area is that
the underlying ideas of CIT can be naturally incorporated into the
study of Killing tensors defined in pseudoRiemannian spaces of
constant curvature. The resulting theory shares many of the same
essential features with CIT.
I target to review some of the results obtained so far. The project
was initiated in a joint work with Ray McLenaghan and Dennis The in
2001.
 DENNIS THE, McGill University
Invariants of Killing tensors and their applications

We present an analysis of Killing tensors in the Euclidean plane using
the moving frame method and derive invariants under the action of the
isometry group. As an application, we demonstrate how these
invariants are useful in the classification of the separable
coordinate systems for the HamiltonJacobi equation.

